Question

Prove by method of induction for all mEN
8+17+26+ .........+(9n-1)=n/2(9n+7)​

Answer 1

Answer:

n = 1

$s = 8 = \frac{1(9.1 + 7)}{2} = 8$

So it's true for n=1

Assume that it's true for n = k and if it holds for k+1 also we are done .

$8+ 17 + ... + (9k - 1) = \frac{k(9k + 7)}{2} \\ \\ add\: \: (k + 1)th \: \: term \: \: both \: \: sides \\ \: \: which \: \: is \: \: 9(k + 1) - 1=9k + 8 \\ \\ 8 + 17 + ..... + (9(k + 1) - 1) \\ \\ = \frac{k(9k + 7)}{2} + 9k +8\\ \\ = \frac{ 9 {k}^{2} + 7k + 18k + 16}{2} \\ \\ = \frac{9 {k}^{2} + 25k + 16 }{2} \\ \\ = \frac{(k + 1)(9k + 16)}{2} \\ \\ = \frac{(k + 1)(9(k + 1) + 7)}{2}$

So it is true for n = k+1

So it will also be true for all nEN .