4. Prove the following statement by mathematical induction: 1 1 + 3-4 + 2.3 1 1.2 + + 1 n(n+1) - n n+1 for all integers n ≥ 1.
4. Prove the following statement by mathematical induction: 1 1 + 3-4 + 2.3 1 1.2 + + 1 n(n+1) - n n+1 for all integers n ≥ 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
4. Prove the following statement by mathematical induction:
\[
\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \cdots + \frac{1}{n(n+1)} = \frac{n}{n+1}
\]
for all integers \( n \geq 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb045dfa-2c18-4fbf-8578-ccec796e086b%2Fb26fc482-c87b-47fb-bb8c-5939508d0c3b%2Fy084v7_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
4. Prove the following statement by mathematical induction:
\[
\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \cdots + \frac{1}{n(n+1)} = \frac{n}{n+1}
\]
for all integers \( n \geq 1 \).
Expert Solution
Step 1
Mathematical induction is a method for proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), ... all hold.
Induction method involves two steps, One, that the statement is true for
n=1
and say
n=2.
Two, we assume that it is true for
n=k
and prove that if it is true for
n=k
, then it is also true for
n=k+1.
Given: 1/(1 * 2) + 1/(2 * 3) + 1/(3 * 4) +...+ 1/n(n+1) = n /n+1
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