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Chapter 7: Q. 66 (page 641)

Use the principle of mathematical induction to prove that if ak+1<akrfor every k 鈮� N, then aN+n<aNrn. Proving this implication completes our proof of the ratio test.

Short Answer

Expert verified

Hence, proved.

Step by step solution

01

Step 1. Given Information.

Givenak+1<akrforeveryk猢�N.

02

Step 2. Proof.

Let suppose it is true for k=N, then

aN+1<aNr.

For k=N+1, we get:

aN+2<aN+1r,aN+2<(aNr)raN+2<aNr2.

The result is true for k=N.

Assume that the result is true for k=N+n-1,

aN+n<aNrn.

Now we have to prove it for k=N+n.

03

Step 3. Proof part 2.

Fork=N+n,aN+n+1<aN+nr,aN+n+1<(aNrn)r,fromstep2.aN+n+1<aNrn+1,Hence,theresultistruefork=N+n.Hencebytheprincipleofmathematicalinduction,resultistrueforeverypositivenumbersk猢�N.

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