418531 ภาคต้น 2552/โจทย์ปัญหาการพิสูจน์ II/เฉลยข้อ 5

ข้อย่อย 1

(Base Case) n มีค่าเท่ากับ 1 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 381026770
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:44 GMT


Sensitive client information

IP address: 158.108.32.49

$f_{1}^{2}=1=1\cdot 1=f_{1}f_{2}$

(Induction Case) ให้ n เป็นจำนวนเต็มบวก และสมมติให้สมการในโจทย์็เป็นจริง 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 384532813 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:44 GMT Sensitive client information IP address: 158.108.32.49 $f_{1}^{2}+f_{1}^{2}+\dotsb +f_{n+1}^{2}\,$	$=\,$	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 365210727 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:45 GMT Sensitive client information IP address: 158.108.32.49 $(f_{1}^{2}+f_{1}^{2}+\dotsb +f_{n}^{2})+f_{n+1}^{2}$
	$=\,$	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 341831740 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:45 GMT Sensitive client information IP address: 158.108.32.49 $f_{n}f_{n+1}+f_{n+1}^{2}$
	$=\,$	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 350312190 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:46 GMT Sensitive client information IP address: 158.108.32.49 $f_{n+1}(f_{n}+f_{n+1})\,$
	$=\,$	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 378439761 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:46 GMT Sensitive client information IP address: 158.108.32.49 $f_{n+1}f_{n+2}\,$

ดังนั้นเราสามารถสรุปได้ว่าสมการในโจทย์เป็นจริงสำหรับจำนวนเต็มบวก n ทุกจำนวน

ข้อย่อย 2

(Base Case) n มีค่าเท่ากับ 1 และเราได้ว่า

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 382239054
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:47 GMT


Sensitive client information

IP address: 158.108.32.49

$f_{1}=1=f_{2}\,$

(Induction Case) สมมติให้ n เป็นจำนวนเต็มบวก 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 383255070 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:47 GMT Sensitive client information IP address: 158.108.32.49 $f_{1}+f_{3}+\dotsb +f_{2n+1}\,$	$=\,$	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 372349097 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:47 GMT Sensitive client information IP address: 158.108.32.49 $(f_{1}+f_{3}+\dotsb +f_{2n-1})+f_{2n+1}$
	$=\,$	$f_{2n}+f_{2n+1}\,$
	$=\,$	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 369862717 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:49 GMT Sensitive client information IP address: 158.108.32.49 $f_{2n+2}\,$

ดังนั้นเราสามารถสรุปได้ว่าสมการในโจทย์เป็นจริงสำหรับจำนวนเต็มบวก n ทุกจำนวน

ข้อย่อย 3

(Base Case) n มีค่าเท่ากับ 1 และเราได้ว่า

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 374378746
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:49 GMT


Sensitive client information

IP address: 158.108.32.49

$f_{1+1}f_{1-1}-f_{1}^{2}=0\cdot 1-1=-1=(-1)^{1}\,$

(Induction Case) สมมติให้ n เป็นจำนวนเต็มบวก 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 382042651 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:49 GMT Sensitive client information IP address: 158.108.32.49 $f_{n+2}f_{n}-f_{n+1}^{2}\,$	$=\,$	$(f_{n+1}+f_{n})f_{n}-f_{n+1}(f_{n}+f{n-1})\,$
	$=\,$	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 384565459 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:50 GMT Sensitive client information IP address: 158.108.32.49 $f_{n+1}f_{n}+f_{n}^{2}-f_{n+1}f_{n}-f_{n+1}f_{n-1}\,$
	$=\,$	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 369046175 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Fri, 23 Jan 2026 18:15:51 GMT Sensitive client information IP address: 158.108.32.49 $-(f_{n+1}f_{n-1}-f_{n}^{2})\,$

ดังนั้นเราสามารถสรุปได้ว่าสมการในโจทย์เป็นจริงสำหรับจำนวนเต็มบวก n ทุกจำนวน