Question

★ Maths Problem !!

1: Find the sum upto nth term of the series :

• $\boxed{\sf{\red{ 4 \:+\: 44 \:+\: 444 \:+\: . . . . . . \:+\: nth\: term }}}$

2: If $\sf{ 1 \:+\: \dfrac{1}{10} \:+\: \dfrac{1}{100} \:+\: . . . . . . . \infty }$ , then find the sum of the series.

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Answer 1

Answer:☝☝

Hope it helps....

Answer 2

Basic Concept :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first n terms of an geometric sequence is,

$\boxed{\red{\bf\:S_n \: = \: \dfrac{a \: ( \: {r}^{n} \: - \: 1) }{r \: - \: 1 \:} \: if \: r \ne \: 1}}$

and

$\boxed{\red{\bf\:S_n \: = \: n \times a, \: \: if \: r \: = \: 1}}$

and

$\boxed{\red{\bf\:S_ \infty \: = \: \dfrac{a}{1 - r} \: \: \: provided \: that \: |r| < 1}}$

Wʜᴇʀᴇ,

• Sₙ is the sum of first n terms.

• a is the first term of the sequence.

• n is the no. of terms.

• r is the common ratio.

Let's solve the problem!!

1. Sum to 'n' terms :-

$\sf \: 4 + 44 + 444 + - - - - - - n \: terms$

$\: \sf \:= \: \: \sf \: 4(1+ 11 + 111 + - - - - - - n \: terms)$

$\: \sf \:= \: \: \sf \: \dfrac{4}{9} (9+99 +999 + - - - - - - n \: terms)$

$\sf \:=\sf \: \dfrac{4}{9} ((10 - 1)+(100 - 1) +(1000 - 1) + - - - n \: terms)$

$\sf \: = \dfrac{4}{9}\bigg((10 + 100 + 100 + - - n \: terms) - (1 + 1 + 1 + - - n \: terms \bigg)$

$\sf \: = \dfrac{4}{9}\bigg(\dfrac{10( {10}^{n} - 1) }{10 - 1} - n \times 1 \bigg) \: \{ \because a = 10,r = 10 \}$

$\sf \because \: S_n \: = \: \dfrac{a \: ( \: {r}^{n} \: - \: 1) }{r \: - \: 1},r \ne1 \: \: and \: \: S_n = n \times a,r = 1$

$\sf \: = \dfrac{4}{9}\bigg(\dfrac{10( {10}^{n} - 1) }{9} - n \bigg)$

$\sf \: = \dfrac{4}{9}\bigg(\dfrac{{10}^{n + 1} - 10 - 9n}{9} \bigg)$

$\sf \: = \dfrac{4}{81}\bigg({10}^{n + 1} - 10 - 9n\bigg)$

Question :- 2

Sum up to infinity :-

$\sf \: 1 + \dfrac{1}{10} + \dfrac{1}{100} + - - - \infty$

Here,

$\rm :\longmapsto\:a = 1 \: \: \: \: and \: r = \dfrac{1}{10}$

So,

Sum of series is given by

$\rm :\longmapsto\:S_ \infty \: = \: \dfrac{a}{1 - r}$

$\rm :\longmapsto\:S_ \infty \: = \: \dfrac{1}{1 - \dfrac{1}{10} }$

$\rm :\longmapsto\:S_ \infty \: = \: \dfrac{1}{ \: \: \: \dfrac{10 - 1}{10} \: \: \: \: }$

$\rm :\longmapsto\:S_ \infty \: = \: \dfrac{1}{ \: \: \: \dfrac{9}{10} \: \: \: \: }$

$\rm :\longmapsto\:S_ \infty \: = \dfrac{10}{9}$