Prove by induction? Thanks :)

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Answer 1 · 2018-03-26T13:37:08

Assume that $3n+5<5^n$ is true for some $n in NN$ . Then

$3(n+1)+5 =(3n+5)+3<5^n+3<5^n times (1+3/5^n)<5^ntimes 5$

where the last inequality follows from $1+3/5^n<1+3<5$ for all $n in NN$ .

This means that
$3(n+1)+5<5^(n+1)$

Thus, if the statement is true for some natural number $n$ , it is true for all subsequent natural numbers.

Now, for $n=2$ , we have

$3n+5=11$

and

$5^n=25$

and so the inequality is true for $n=2$ . Hence it is true for all natural numbers $ge 2$ .

Note that for $n=1$ , we have

$3n+5= 8 > 5=5^1$

so that the inequality does not hold for $n=1$ !