Evaluate : 9 + 99 + 999 + …….. upto n terms.

Given,

⇒ 9 + 99 + 999 + …….. upto n terms.

⇒ (10 - 1) + (100 - 1) + (1000 - 1) + ……… upto n terms

⇒ (10 + 100 + 1000 + ……. upto n terms) + (-1 + -1 + -1 + ……… upto n terms)

⇒ (10 + 10² + 10³ + ……. upto n terms) + (-n)

(10 + 10² + 10³ + ……. upto n terms) is a G.P. with first term (a) = 10 and common ratio (r) = 10.

By formula,

Sum of G.P. = $\dfrac{a(r^n - 1)}{r - 1}$

$\Rightarrow \dfrac{10(10^n - 1)}{10 - 1} + (-n) \\[1em] \Rightarrow \dfrac{10(10^n - 1)}{9} - n.$

Hence, 9 + 99 + 999 + …….. upto n terms = $\dfrac{10(10^n - 1)}{9} - n.$