Why is it that we need to change n to k in proving? What is the idea or reason behind that?
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
Problem 1RQ
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Question
Why is it that we need to change n to k in proving? What is the idea or reason behind that?
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73fc84ba-31f2-4c34-9d71-484f34e776ea%2F348bbce2-601c-4d3e-a51c-db4bb2e773c3%2Fdjmxyz_processed.png&w=3840&q=75)
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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