Question

asked May 14, 2020 1.5k views

1 Answer

Toreyhickman · Answer 1 · 2020-05-21T06:09:01+0000

For making mathematical induction, we need:

An
n_0 for which the relation holds true

if its true for
n_i , then, is true for
n_(i+1)

the relationship is not true for 1 or 2

1^3-1 = 0 < 4*1

2^3-1 = 8 -1 = 7 < 4*2 = 8

but, is true for 3

3^3-1 = 27 -1 = 26 > 4*3 = 12

lets say that the relationship is true for n, this is

$n^3 -1 \ge 4 n$

lets add 4 on each side, this is

$n^3 -1 + 4 \ge 4 n + 4$

$n^3 + 3 \ge 4 (n + 1)$

now

(n+1)^3 = n^3 +3 n^2 + 3 n + 1

$(n+1)^3 \ge n^3 + 3 n$

if
$n \ge 1$ then
$3 n \ge 3$ , so

$(n+1)^3 \ge n^3 + 3 n \ge n^3 + 3$

$(n+1)^3 \ge  n^3 + 3 \ge 4 (n + 1)$

$(n+1)^3  \ge 4 (n + 1)$

and this is what we were looking for!

So, for any natural equal or greater than 3, the relationship is true.

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