a) Find a formula for 1/1\cdot2 + 1/2\cdot3 + · · · + 1/n(n+1) by examinin

iohanetc

iohanetc

Answered question

2021-09-17

a) Find a formula for 112+123+···+1n(n+1) by examining the values of this expression for small values of n. b) Prove the formula you conjectured in part (a).

Answer & Explanation

nitruraviX

nitruraviX

Skilled2021-09-18Added 101 answers

a) Given:
f(n)=112+123+134++1n(n+1)
We will evaluate the first few values of n and look for a pattern:
n=1 112=12
n=2 112+123=12+16=36+16=46=23
n=3 112+123+134=12+16+112=612+212+112=912=34
We then note a pattern
f(n)=nn+1
b) To proof: 112+123+134++1n(n+1)=nn+1 for every positive integer.
Proof by induction
Let P(n) be 112+123+134++1n(n+1)=nn+1
Basis step n=1
112=12
11+1=12
We then note P(1) is true, as both sides of the equations is equal to 12
Induction step Let P(k) be true.
112+123+134++1k(k+1)=kk+1
We need to prove that P(k+1) is also true.

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