Use the Principle of Mathematical Induction (PMI) to prove the following for all natural numbers n. 1+2+3+ ·· +n= n(n + 1) 2
Use the Principle of Mathematical Induction (PMI) to prove the following for all natural numbers n. 1+2+3+ ·· +n= n(n + 1) 2
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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Transcribed Image Text:Use the Principle of Mathematical Induction (PMI) to prove the
following for all natural numbers n.
1+2+3+...+n=
n(n+1)
2
![. Use the Principle of Mathematical Induction (PMI) to prove the
following for all natural numbers n.
• P(1) is
3
P(n)
3+ 11 +19++ (8n — 5) = 4n² — n (neN) Pin)
true
because
4(1)²_1=3 ✓
ころ
is true for some neN.
3 +11 +19+ - • -+ (8n −5)]=4n²_n_P(n)
•Assume P(n)
Need to prove Pin+1)
[3+11+19+---+ (8n - 5] + [8(n+1)-5] pon+1)
=4[n+1]²_[n+i]
Proof of pintl):
3+1+19+...+ (8n_5)+(8n+3)
= 4n²_n +8n-3
=4n²+7n-3
Note: 4[n+1]-[n+1] =4[n²+²n +i]-[n+i]
3
=4n² +8n++-n-1
=4n²+7+3
This proves P(n+l).
•
PMI implies (EN) P(n) is true.
A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd49a5070-9d99-4a2f-a144-be47e7ddead5%2Fda27416d-a80e-41b9-8276-5504262e4843%2Fsut7ayq_processed.png&w=3840&q=75)
Transcribed Image Text:. Use the Principle of Mathematical Induction (PMI) to prove the
following for all natural numbers n.
• P(1) is
3
P(n)
3+ 11 +19++ (8n — 5) = 4n² — n (neN) Pin)
true
because
4(1)²_1=3 ✓
ころ
is true for some neN.
3 +11 +19+ - • -+ (8n −5)]=4n²_n_P(n)
•Assume P(n)
Need to prove Pin+1)
[3+11+19+---+ (8n - 5] + [8(n+1)-5] pon+1)
=4[n+1]²_[n+i]
Proof of pintl):
3+1+19+...+ (8n_5)+(8n+3)
= 4n²_n +8n-3
=4n²+7n-3
Note: 4[n+1]-[n+1] =4[n²+²n +i]-[n+i]
3
=4n² +8n++-n-1
=4n²+7+3
This proves P(n+l).
•
PMI implies (EN) P(n) is true.
A
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