เฉลย

ข้อย่อย 1

สูตรคือ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 518362809
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:19 GMT


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${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n}}}={\frac {2^{n}-1}{2^{n}}}$

base case: ให้

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 520296327
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:20 GMT


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$n=1$

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 518362813
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:20 GMT


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${\frac {1}{2^{1}}}={\frac {2^{1}-1}{2^{1}}}={\frac {1}{2}}$ เป็นจริง

inductive step: inductive hypothesis คือสมมติให้ p(n) คือ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 480402766
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:20 GMT


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${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n}}}={\frac {2^{n}-1}{2^{n}}}$ เป็นจริง ต้องการพิสูจน์ว่า p(n+1) คือ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 521541024
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:20 GMT


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${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n+1}}}={\frac {2^{n+1}-1}{2^{n+1}}}$ เป็นจริงด้วย

จากที่สมมติไว้คือ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 497896000
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:21 GMT


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${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n}}}={\frac {2^{n}-1}{2^{n}}}$

บวกทั้งสองข้างของสมการด้วย

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 494879072
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:21 GMT


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${\frac {1}{2^{n+1}}}$

จะได้ ${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n}}}+{\frac {1}{2^{n+1}}}={\frac {2^{n}-1}{2^{n}}}+{\frac {1}{2^{n+1}}}$

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 533595467
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:22 GMT


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$={\frac {2}{2}}.{\frac {2^{n}-1}{2^{n}}}+{\frac {1}{2^{n+1}}}$

$={\frac {2^{n+1}-2}{2^{n+1}}}+{\frac {1}{2^{n+1}}}$

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 510730039
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:22 GMT


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$={\frac {2^{n+1}-1}{2^{n+1}}}$

ดังนั้นจึงสรุปได้ว่า ${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n}}}={\frac {2^{n}-1}{2^{n}}}$

ข้อย่อย 2

สูตรคือ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 468579499
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:23 GMT


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${\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\cdots +{\frac {1}{n(n+1)}}={\frac {n}{n+1}}$

base case: ให้

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 522687238
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:23 GMT


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$n=1$

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 505229354
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:23 GMT


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${\frac {1}{1(1+1)}}={\frac {1}{1+1}}={\frac {1}{2}}$ เป็นจริง

inductive step: inductive hypothesis คือสมมติให้ p(n)คือ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 500446300
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:24 GMT


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${\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\cdots +{\frac {1}{n(n+1)}}={\frac {n}{n+1}}$ เป็นจริง ต้องการพิสูจน์ว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 519935770
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:24 GMT


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${\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\cdots +{\frac {1}{(n+1)(n+2)}}={\frac {n+1}{n+2}}$ เป็นจริงด้วย

จากที่สมมติไว้คือ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 533103930
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:24 GMT


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${\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\cdots +{\frac {1}{n(n+1)}}={\frac {n}{n+1}}$

บวก

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 520066374
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:24 GMT


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${\frac {1}{(n+1)(n+2)}}$ ทั้งสองข้างของสมการ

จะได้

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 512467004
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:25 GMT


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${\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\cdots +{\frac {1}{n(n+1)}}+{\frac {1}{(n+1)(n+2)}}={\frac {n}{n+1}}+{\frac {1}{(n+1)(n+2)}}$

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 517249914
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:25 GMT


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$={\frac {n}{(n+1)}}.{\frac {(n+2)}{(n+2)}}+{\frac {1}{(n+1)(n+2)}}$

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 527207716
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:25 GMT


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$={\frac {(n+1)(n+1)}{(n+1)(n+2)}}$

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 474440738
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:25 GMT


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$={\frac {(n+1)}{(n+2)}}$

ดังนั้นเราจึงสรุปได้ว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 531728689
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:26 GMT


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${\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\cdots +{\frac {1}{n(n+1)}}={\frac {n}{n+1}}$ เป็นจริง

ข้อย่อย 3

(Base Case) เนื่องจาก

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 488003595
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:26 GMT


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$n=0\,$ เราได้ว่า $1=(2\cdot 0+1)^{2}=(0+1)(2\cdot 0+1)(2\cdot 0+3)/3\,$

(Induction Case) สมมติให้ n เป็นจำนวนเต็มที่มีค่ามากกว่าหรือเท่ากับ 0 และสมมติให้สมการในโจทย์เป็นจริง เราได้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 529107626 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:27 GMT Sensitive client information IP address: 158.108.32.49 $1^{2}+3^{2}+\dotsb +(2(n+1)+1)^{2}\,$	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 480435271 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:27 GMT Sensitive client information IP address: 158.108.32.49 $(1^{2}+3^{2}+\dotsb +(2n+1)^{2})+(2n+3)^{2}\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 521213076 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:28 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {(n+1)(2n+1)(2n+3)}{3}}+(2n+3)^{2}\,$
	$=\,$	$(2n+3){\bigg [}{\frac {(n+1)(2n+1)}{3}}+(2n+3){\bigg ]}\,$
	$=\,$	$(2n+3){\bigg [}{\frac {2n^{2}+3n+1+3(2n+3)}{3}}{\bigg ]}\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 508113793 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:30 GMT Sensitive client information IP address: 158.108.32.49 $(2n+3){\bigg [}{\frac {2n^{2}+9n+10}{3}}{\bigg ]}\,$
	$=\,$	$(2n+3){\frac {(n+2)(2n+5)}{3}}\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 513808677 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:31 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {((n+1)+1)((2(n+1)+1)(2(n+1)+3)}{3}}\,$

ฉะนั้นเราจึงสามารถสรุปได้ว่าสมการในโจทย์เป็นจริงสำหรับจำนวนเต็ม n ที่ไม่เป็นลบทุกจำนวน

ข้อย่อย 4

base case: คือ n=5 แทนค่าจะได้

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 523998221
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:32 GMT


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$2^{5}>5^{2}$ เป็นจริง

inductive step: inductive hypothesis คือ สมมติให้ p(n) คือ $2^{n}>n^{2}$ เป็นจริง ต้องการแสดงว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 507226603
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:32 GMT


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$2^{n+1}>(n+1)^{2}$ เป็นจริงด้วย

จากที่สมมติ $2^{n}>n^{2}$

คูณ 2 ทั้งสองข้างของสมการจะได้

$2.2^{n}>2.n^{2}$

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 491505268
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:33 GMT


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$2^{n+1}>n^{2}+n^{2}$

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 521311548
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:33 GMT


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$\geq n^{2}+5n$ เนื่องจาก n>4

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 501792311
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:34 GMT


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$\geq n^{2}+2n+1$

$=(n+1)^{2}$

ดังนั้นเราจึงสรุปได้ว่า $2^{n}>n^{2}$ เป็นจริง เมื่อ n เป็นจำนวนเต็มบวกที่มีค่ามากกว่า 4

ข้อย่อย 5

(Base Case) n มีค่าเท่ากับ 2 และเราได้ว่า $1+1/4=5/4<3/2=2-1/2\,$

(Induction Case) สมมติให้ n เป็นจำนวนเต็มบวกที่มีค่ามากกว่า 2 และให้สมการในโจทย์เป็นจริง เราได้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 500383442 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:35 GMT Sensitive client information IP address: 158.108.32.49 $1+{\frac {1}{4}}+\dotsb +{\frac {1}{n+1}}^{2}\,$	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 530025038 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:35 GMT Sensitive client information IP address: 158.108.32.49 $(1+{\frac {1}{4}}+\dotsb +{\frac {1}{n^{2}}})+{\frac {1}{(n+1)^{n}}}\,$
	$<\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 517250136 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:36 GMT Sensitive client information IP address: 158.108.32.49 $2-{\frac {1}{n}}+{\frac {1}{(n+1)^{2}}}\,$
	$<\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 513482352 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:37 GMT Sensitive client information IP address: 158.108.32.49 $2-{\frac {1}{n}}+{\frac {1}{n(n+1)}}\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 478796376 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:37 GMT Sensitive client information IP address: 158.108.32.49 $2-{\frac {1}{n}}{\Bigg [}1-{\frac {1}{n+1}}{\Bigg ]}\,$
	$=\,$	$2-{\frac {1}{n}}{\frac {n}{n+1}}\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 503951889 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:38 GMT Sensitive client information IP address: 158.108.32.49 $2-{\frac {1}{n+1}}\,$

ดังนั้นเราสามารถสรุปได้ว่าสมการในโจทย์เป็นจริงสำหรับจำนวนจริง

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 527894658
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:39 GMT


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$n>1$ ทุกจำนวน

ข้อย่อย 6

(Base Case) n มีค่าเท่ากับ 0 และเราได้ว่า $0^{3}-0=0$ ซึ่งหารด้วย 6 ได้ลงตัว

(Induction Case) สมมติว่า n เป็นจำนวนเต็มที่ไม่เป็นลบ และสมมติให้ $n^{3}-n$ หารด้วย 6 ลงตัว

พิจารณาค่า $(n+1)^{3}-(n+1)=n^{3}+3n^{2}+3n+1-n-1=(n^{3}-n)+3n^{2}+3n=(n^{3}-n)+3n(n+1)$

เราได้ว่า 6 หาร

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 515775416
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:40 GMT


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$3n(n+1)$ ลงตัวเนื่องจาก 3 หาร

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 485120439
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:41 GMT


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$3n(n+1)$ ลงตัว นอกจากนี้ 2 ยังหาร

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 498416627
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:41 GMT


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$3n(n+1)$ ลงตัวเนื่องจากในค่า $n$ และ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 523178017
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:41 GMT


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$n+1$ ลงตัว จะต้องมีสักตัวที่เป็นจำนวนคู่

เนื่องจาก 6 หารทั้ง $n^{3}-n$ และ $3n(n+1)$ ลงตัว เราจึงได้ว่า 6 หาร

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 517544691
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:42 GMT


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$(n^{3}-n)+3n(n+1)=(n+1)^{3}-(n+1)$ ลงตัวด้วย

ดังนั้นเราจึงสรุปได้ว่า 6 หาร $n^{3}-n$ ลงตัวสำหรับจำนวนเต็ม n ที่ไม่เป็นลบทุกจำนวน

ข้อย่อย 7

ก่อนเราจะทำการพิสูจน์ข้อความในโจทย์ เราจะทำการพิสูจน์ lemma ต่อไปนี้

lemma: ให้

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 479652540
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:43 GMT


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$A\,$, $B\,$,

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Too many requests (f061ab2)

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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:43 GMT


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$C\,$ และ

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Request served via cp5021 cp5021, Varnish XID 500644804
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:44 GMT


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$D\,$ เป็นเซตใดๆ ที่ $A\subseteq C\,$ และ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 502312215
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:44 GMT


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$B\subseteq D\,$ แล้ว

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 462484216
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:45 GMT


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$A\cap B\subseteq C\cap D$

พิสูจน์ (lemma): ให้ x เป็นค่าใดๆ สมมติให้

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 494094718
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:45 GMT


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$x\in A\cap B$ เราได้ว่า และ

เนื่องจาก $A\subseteq C$ และ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 508080556
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:46 GMT


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$B\subseteq D$ เราได้ว่า และ

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 532482046
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:46 GMT


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$x\in D$ ด้วย ดังนั้น

เนื่องจาก x เป็นค่าใดๆ เราจึงสามารถสรุปได้ว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 521703843
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:46 GMT


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$\forall x,[x\in A\cap B\rightarrow x\in C\cap D]$ ฉะนั้น

พิสูจน์ (โจทย์)

(Base Case) n มีค่าเท่ากับ 1 ในกรณีนี้เราได้ว่า $\bigcap _{i=1}^{1}A_{i}=A\subseteq B=\bigcap _{i=1}^{1}B_{i}\,$

(Induction Case) ให้ n เป็นจำนวนเต็มบวก และสมมติให้ข้อความที่โจทย์ต้องการพิสูจน์เป็นจริง

เราได้ว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 525899661
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:47 GMT


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$\bigcap _{i=1}^{n+1}A_{i}=A_{n+1}\cap \bigcap _{i=1}^{n}A_{i}$ และ

โจทย์กำหนดว่า และจาำกสมมติฐานเราได้ว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 488786088
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:47 GMT


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$\bigcap _{i=1}^{n}A_{i}=A\bigcap _{i=1}^{n}B_{i}\,$ ฉะนั้นด้วย lemma เราได้ว่า

ฉะนั้นเราจึงสรุปได้ว่าข้อความในโจทย์เป็นจริงสำหรับจำนวนเต็มบวก n ทุกค่า

ข้อย่อย 8

(Base Case) n มีค่าเท่ากับ 1 และเราจะได้ว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 517611975
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:48 GMT


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$1\cdot 2^{1-1}=1=(1-1)2^{1}+1$

(Induction Case) ให้ n เป็นจำนวนเต็มที่มีค่ามากกว่า 0 และสมมติให้สมการในโจทย์เป็นจริง ได้ว่า

$1\cdot 2^{0}+2\cdot 2^{1}+\dotsb +(n+1)\cdot 2^{n}\,$		Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 468379377 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:48 GMT Sensitive client information IP address: 158.108.32.49 $(1\cdot 2^{0}+2\cdot 2^{1}+\dotsb +n\cdot 2^{n-1})+(n+1)\cdot 2^{n}\,$
	$=\,$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 473813894 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:01:16 GMT Sensitive client information IP address: 158.108.32.49 $(n-1)2^{n}+1+(n+1)\cdot 2^{n}\,$
	$=\,$
	$=\,$

ดังนั้นเราสรุปได้ว่าสมการเป็นจริงสำหรับจำนวนเต็มบวก n ทุกจำนวน

ข้อ 9

(Base Case) n มีค่าเท่ากับ 1 และเราได้ว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 521180085
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:49 GMT


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$4^{1+1}+5^{2\cdot 1-1}=16+5=21$ ซึ่งหารด้วย 21 ลงตัว

(Induction Case) ให้ n เป็นจำนวนเต็มบวก และสมมติให้ หารด้วย 21 ลงตัว

เราได้ว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 491340630
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:49 GMT


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$4^{n+2}+5^{2n+1}=4\cdot 4^{n+1}+25\cdot 5^{2n-1}=(25-21)4^{n+1}+25\cdot 5^{2n-1}=25(4^{n+1}+5^{2n-1})+21\cdot 4^{n+1}$

จากสมมติฐาน เราได้ว่า 21 หาร ลงตัว ดังนั้นมันจึงหาร

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 478436933
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:50 GMT


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$25(4^{n+1}+5^{2n-1})$ ลงตัว และเนื่องจาก 21 หาร ลงตัว เราจึงได้ว่า 21 หาร

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 484265403
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:01:21 GMT


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$25(4^{n+1}+5^{2n-1})+21\cdot 4^{n+1}=4^{n+2}+5^{2n+1}$

ฉะนั้นเราจึงสามารถสรุปได้ว่า 21 หาร ลงตัวสำหรับจำนวนเต็มบวก n ทุกจำนวน

ข้อ 10

lemma: สำหรับจำนวนเต็มบวก n ใดๆ

พิสูจน์ (lemma): เราได้ว่า

Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 515119974 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:50 GMT Sensitive client information IP address: 158.108.32.49 ${\sqrt {n+2}}+{\sqrt {n+1}}$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 458095904 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:50 GMT Sensitive client information IP address: 158.108.32.49 $>\,$
Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 502969010 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:51 GMT Sensitive client information IP address: 158.108.32.49 $({\sqrt {n+2}}+{\sqrt {n+1}}){\frac {{\sqrt {n+2}}-{\sqrt {n+1}}}{{\sqrt {n+2}}-{\sqrt {n+1}}}}$
Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 519312983 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:51 GMT Sensitive client information IP address: 158.108.32.49 ${\frac {1}{{\sqrt {n+2}}-{\sqrt {n+1}}}}$	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 478666453 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:51 GMT Sensitive client information IP address: 158.108.32.49 $>\,$
${\frac {1}{\sqrt {n+1}}}$
	Error Too many requests (f061ab2) If you report this error to the Wikimedia System Administrators, please include the details below. Request served via cp5021 cp5021, Varnish XID 505949437 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:52 GMT Sensitive client information IP address: 158.108.32.49 $>\,$	$2{\sqrt {n+2}}$

พิสูจน์ (โจทย์)

(Base Case) n มีค่าเท่ากับ 1 เราได้ว่า

Error

Too many requests (f061ab2)

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Request served via cp5021 cp5021, Varnish XID 496191389
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Error: 429, Too many requests (f061ab2) at Sun, 15 Mar 2026 12:00:52 GMT


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$1>0.82842712474619029\dotsb =2({\sqrt {1+1}}-1)\,$

(Induction Case) ให้ n เป็นจำนวนเต็มบวก และสมมติให้อสมการในโจทย์เป็นจริง เราได้ว่า

ดังนั้นเราสามารถสรุปได้ว่าอสมการในโจทย์เป็นจริงสำหรับจำนวนเต็มบวก n ทุกตัว