Question

Sum to n terms the series :-
(i) 7 + 77 + 777 + ....
(ii) 0.7 + 0.77 + 0.777 + ....
​

Answer 1

Sum to n terms the series :-

Answer 2

★ Concept

The above question is based on the concept of Sum of 'n' terms of a Geometric Series. Before starting with solution have a look on the derivation of the basic formula for the Geometric Sum so as to compare it with the given series.

Derivation

$\sf Consider \: the \: series : \red{ a + ar + a {r}^{2} + .... + a {r}^{n - 1} }$

Let Sn denote the sum of 'n' terms of this series. Write the series and subtract from it, term by term, the product of 'r' and the series as shown :-

$\sf \red{S_n} = a + ar + a {r}^{2} + .... + a {r}^{n - 1}$

$\sf r. S_n = ar + a {r}^{2} + .... + a {r}^{n - 1} + a {r}^{n - 2} + a {r}^{n}$

Subtracting, we'll get

$\sf \: S_n - r. S_n = a - a {r}^{n} \implies S_n(1 - r)$

$\sf S_n = \dfrac{a(1 - {r}^{n})}{1 - r} \quad \: ( \red{r \cancel{=} 1})$

$\rule{190pt}{1pt}$

Let's proceed with calculation !!

$\sf \red{1)} \: 7 + 77 + 777 + ...$

$\sf \: S_n = 7 + 77 + 777 + .... to \: \red{n} \: terms.$

$\sf = 7[1 + 11 + 111 + ...to \: \red{n} \: terms]$

$\sf =\dfrac{7}{9}[9 + 99 + 999 + ... \: to \: n \: terms]$

$\sf = \dfrac{7}{9}[(10 - 1) + (100 - 1) + (1000 - 1) + ... \: to \: n \: terms]$

$\sf = \dfrac{7}{9} [(10 + {10}^{2} + {10}^{3} + ... \: to \: n \: terms)- (1 + 1 + 1 + ... \: to \: n \: terms)]$

$\sf = \dfrac{7}{9} \huge{[} \footnotesize \sf\dfrac{10. ({10}^{n} - 10)}{81} \huge{ ]}$

$: \implies \underline{\boxed{ \sf\dfrac{7( {10}^{n + 1} - 10)}{81} - \dfrac{7}{9}n}}$

$\sf \red{2)} \: 0.7 + 0.77 + 0.777 + ...$

$\sf S_n = \: 0.7 + 0.77 + 0.777 + ... to \: \red n \: terms$

$\sf = 7[0.1 + 0.11 + 0.111 + ...to \: \red{n} \: terms]$

$\sf = \dfrac{7}{9}[0.9 + 0.99 + 0.999 + ... \: to \: n \: terms]$

$\sf =\dfrac{7}{9}[(1 - 0.1) + (1 - 0.01) + (1- 0.001) + ... \: to \: n \: terms]$

$\sf = \dfrac{7}{9} \huge{[} \footnotesize \sf \: n - \dfrac{0.1(1 - {0.1}^{n})}{1 - 0.1} \huge{ ]}$

$\sf:\implies\underline{ \boxed{ \dfrac{7}{9} \huge{ [ }\footnotesize \sf \: n - \dfrac{1}{9} (1 - \dfrac{1}{ {10}^{n}}) \huge{]}}}$

$\underline{\rule{190pt}{2pt}}$

✧ Note

➠ Refer to the derivation first before starting with the solution. We can write and use the above formula in the form of

$\sf S_n = \dfrac{a({r}^{n}\: -\: 1)}{r - 1} \quad \: ( \red{r \cancel{=} 1})$