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Sequence and Series

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Question : 72 of 86

Marks: +1, -0

1+11+111+. . . . . . . . . . .

n terms.

$[10 n+1-9 n-10]$
$[10 n+1-9 n-10]$
$[10 n+1-9 n-10]$
none of these

Solution:

Concept:

Each term is of the form

\frac{10^k - 1}{9}

.

Explanation:

Write the series:

1 = \frac{10-1}{9}

,

11 = \frac{10^2-1}{9}

,

111 = \frac{10^3-1}{9}

, ..., up to

n

terms.

Sum

S = \frac{1}{9} \left[ (10 + 10^2 + \dots + 10^n) - n \right]

.

The geometric series sum:

10 + 10^2 + \dots + 10^n = \frac{10(10^n - 1)}{9} = \frac{10^{n+1} - 10}{9}

.

Thus

S = \frac{1}{9} \left[ \frac{10^{n+1} - 10}{9} - n \right] = \frac{10^{n+1} - 10 - 9n}{81} = \frac{10^{n+1} - 9n - 10}{81}

.

The options A, B, C all show

10^{n+1} - 9n - 10

without the denominator 81, so none of these is correct.

Answer:

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