Question

plz answer!!!!!!!!!!!! ​

Answer 1

Here's the Solution :

Let

S(n) = 1 + x + x² + x³ +......+x^n-1 =(x^n - 1)/(x - 1)

Taking n = 1 we have

= ( x¹ - 1) / (x -1)

=( x -1)/(x - 1)

= 1

Thus S(n) is true for n = 1

Now , let S(k) = 1 + x +x² +x³+...+x^k-1 = ( x^k - 1)/(x - 1)

Adding x^k on both side we have :

1+x+x²+x³+...+x^(k-1)+x^k = x^k + (x^k - 1)/(x - 1)

S(k) + x^k=[(x- 1)x^k + x^k - 1]/(x-1)

$\frac{(x - 1) {x}^{k} + {x}^{k} - 1}{x - 1} \\ \\ = \frac{ {x}^{k}(x - 1 + 1) - 1 }{x - 1} \\ \\ = \frac{ {x}^{k} \times x - 1 }{x - 1} \\ \\ = \frac{ {x}^{k + 1} - 1 }{x - 1}$

Thus S(n) is also true for n = k+1 .

Therefore , by mathematical induction we prove that S(n) is true for any integer 'n'

Happy Learning