Why is it that we need to change n to k in proving? What is the idea or reason behind that?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Why is it that we need to change n to k in proving? What is the idea or reason behind that?

1. Prove: S₁₁ =1+3+ 5 + 7 + ... + (2n − 1) = n²
Base Step:
S₁ = 1² (261)-1) ² (1) ²
S₁=1=1=1 ✓
Inductive Step: Assume Sk = K²
Prove
Sk+₁ = (K+1) ²
SK+ = Sx +ak+1
k² + 2(K+1)-1
= K² +2K+2-1
= K ²+2K+1
= (K+1) ²
Transcribed Image Text:1. Prove: S₁₁ =1+3+ 5 + 7 + ... + (2n − 1) = n² Base Step: S₁ = 1² (261)-1) ² (1) ² S₁=1=1=1 ✓ Inductive Step: Assume Sk = K² Prove Sk+₁ = (K+1) ² SK+ = Sx +ak+1 k² + 2(K+1)-1 = K² +2K+2-1 = K ²+2K+1 = (K+1) ²
Prove that 1 + 2 + 3 + ... + n =
positive integers.
Verification: n = 1
1(1 + 1)
2
1 =
1 =
1(2)
2
2
2
1 = 1
therefore, true for n = 1
Prove that 1+2+3+...+k+k+1 =
Induction Hypothesis: n=k Assume that 1+2+3+...+ k =
Proof of Induction: n = k + 1
Prove that 1+ 2+ 3+ ... + k +k+1 =
(2)
k(k + 1)
2
+ k + 1
n(n+1)
2
k(k+ 1) +2k + 2
2
k+ 1[(k+ 1) + 1]
2
(k+ 1)(k+2)
2
k² + k + 2k +2
2
for all
k² + 3k + 2
2
k(k+1)
2
Transcribed Image Text:Prove that 1 + 2 + 3 + ... + n = positive integers. Verification: n = 1 1(1 + 1) 2 1 = 1 = 1(2) 2 2 2 1 = 1 therefore, true for n = 1 Prove that 1+2+3+...+k+k+1 = Induction Hypothesis: n=k Assume that 1+2+3+...+ k = Proof of Induction: n = k + 1 Prove that 1+ 2+ 3+ ... + k +k+1 = (2) k(k + 1) 2 + k + 1 n(n+1) 2 k(k+ 1) +2k + 2 2 k+ 1[(k+ 1) + 1] 2 (k+ 1)(k+2) 2 k² + k + 2k +2 2 for all k² + 3k + 2 2 k(k+1) 2
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