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

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

`2+7+12+17+....+(5n-3)=(n(5n-1))/(2)`

`(n(5n-1))/(2)`

when n = 1,

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

`\begin{gathered} 2=(1)/(2)(5(1)-1) \\ 2=(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:

`(n(5n-1))/(2)`