Question

Prove by induction? Thanks :)

Answer 1

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`!