Question

Show that the following is true by mathematical induction for all natural numbers of n2 + 7 + 12 + 17 + __ +(5n-3)= (n(5n-1))/2

Answer 1

Solution

Show that the following is true by mathematical induction for all natural numbers of n

$2+7+12+17+....+(5n-3)=\frac{n(5n-1)}{2}$

$\frac{n(5n-1)}{2}$

when n = 1,

The claim is true for n = 1, a_1 = 2

$\begin{gathered} 2=\frac{1}{2}(5(1)-1) \\ 2=\frac{1}{2}.4 \\ 2=2 \end{gathered}$

Assume the claim is true for n , and look for n+1

$2+7+....+5n-3+5(n+1)-3$

The induction hypothesis is that the first n terms sum up to:

$\frac{n(5n-1)}{2}$