418531 ภาคต้น 2552/โจทย์ปัญหาการพิสูจน์ II/เฉลยข้อ 9

เราจะแสดงว่า

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 963052718
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Feb 2026 21:26:51 GMT


Sensitive client information

IP address: 158.108.32.49

$T(n)=n!+2^{n}\,$ สำหรับจำนวนเต็ม

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 915925020
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Feb 2026 21:26:52 GMT


Sensitive client information

IP address: 158.108.32.49

$n\geq 0\,$ ทุกจำนวน

(Base Case) มีอยู่สามกรณี

• 

  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 962363917
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Feb 2026 21:26:52 GMT


  Sensitive client information

  IP address: 158.108.32.49

  $T_{0}=2=2^{0}+0!\,$
• 

  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 961841920
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Feb 2026 21:26:52 GMT


  Sensitive client information

  IP address: 158.108.32.49

  $T_{1}=3=2+1=2^{1}+1!\,$
• 

  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 942287069
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Feb 2026 21:26:52 GMT


  Sensitive client information

  IP address: 158.108.32.49

  $T_{2}=6=4+2=2^{2}+2!\,$

(Induction Case) ให้ $n\,$ เป็นจำนวนเต็มบวกที่มีค่ามากกว่าหรือเท่ากับ 2 และสมมติให้

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 962102614
Upstream caches: cp5021 int
Error: 429, Too many requests (f061ab2) at Sun, 15 Feb 2026 21:26:53 GMT


Sensitive client information

IP address: 158.108.32.49

$T_{k}=k!+2^{k}\,$ สำหรับจำนวนเต็ม $k\,$ ทุกตัวที่มีค่าตั้งแต่ 0 ถึง $n\,$ เราได้ว่า

$T_{n+1}\,$	$=\,$	$(n+1+4)T_{n}-4(n+1)T_{n-1}+(4n-4)T_{n-2}\,$
	$=\,$	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 957814404 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Feb 2026 21:26:56 GMT Sensitive client information IP address: 158.108.32.49 $(n+1+4)(n!+2^{n})-(4n+4)((n-1)!+2^{n-1})+(4n-4)((n-2)!+2^{n-2})\,$
	$=\,$	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 957616866 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Feb 2026 21:26:56 GMT Sensitive client information IP address: 158.108.32.49 $(n+1)!+4n!+n\cdot 2^{n}+5\cdot 2^{n}-4n!-4(n-1)!-2n\cdot 2^{n}-2\cdot 2^{n}+4(n-1)!+n\cdot 2^{n}-2^{n}\,$
	$=\,$	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 939535505 Upstream caches: cp5021 int Error: 429, Too many requests (f061ab2) at Sun, 15 Feb 2026 21:26:57 GMT Sensitive client information IP address: 158.108.32.49 $(n+1)!+2\cdot 2^{n}=(n+1)!+2^{n+1}\,$

ดังนั้นเราสามารถสรุปได้ว่า สำหรับจำนวนเต็ม ที่มีค่าไม่เป็นลบทุกจำนวน