1 + (1 + x) + (1 + x + x2) + ....... n terms
⇒ Required sum = \(\frac{1}{(1-x)}\) [(1 – x) + (1 – x) (1 + x) + (1 – x) (1 + x + x2) + ..... n terms]
= \(\frac{1}{(1-x)}\) [(1 – x) + (1 – x2) + (1 – x3) + ..... n terms]
= \(\frac{1}{(1-x)}\) [(1 + 1 + 1 + .... n terms) – (x + x2 + x3 + ..... n terms]
= \(\frac{1}{(1-x)}\)\(\bigg[n-\frac{x(1-x^n)}{(1-x)}\bigg]\) \(\bigg(\because{S_n}=\frac{a(1-r^n)}{(1-r)},\text{Here}\,a= x, r=x\bigg)\)
\(\frac{n(1-x)-x(1-x^n)}{(1-x)^2}\).