Sum of n terms of the series $8+88+888+........$ is equal to :

Question

Accepted Answer

To determine the sum of the first $ n $ terms of the series $ 8 + 88 + 888 + \ldots $, we need to understand the pattern of the series. Each term in the series can be expressed in a form that involves powers of 10. Let's break down the series:

1. The first term is $ 8 $.
2. The second term is $ 88 = 8 \times 10 + 8 $.
3. The third term is $ 888 = 8 \times 100 + 8 \times 10 + 8 $.
4. The fourth term is $ 8888 = 8 \times 1000 + 8 \times 100 + 8 \times 10 + 8 $.

From this pattern, we can see that the $ k $-th term can be expressed as:
$$ T_k = 8 \times (10^{k-1} + 10^{k-2} + \ldots + 10^0) $$

This is a geometric series with $ k $ terms, where the first term $ a = 1 $ and the common ratio $ r = 10 $. The sum of this geometric series is:
$$ 10^{k-1} + 10^{k-2} + \ldots + 10^0 = \frac{10^k - 1}{10 - 1} = \frac{10^k - 1}{9} $$

Thus, the $ k $-th term becomes:
$$ T_k = 8 \times \frac{10^k - 1}{9} $$

The sum of the first $ n $ terms of the series is:
$$ S_n = \sum_{k=1}^{n} T_k = \sum_{k=1}^{n} \left( 8 \times \frac{10^k - 1}{9} \right) $$

$$ S_n = \frac{8}{9} \sum_{k=1}^{n} (10^k - 1) $$

$$ S_n = \frac{8}{9} \left( \sum_{k=1}^{n} 10^k - \sum_{k=1}^{n} 1 \right) $$

The sum $ \sum_{k=1}^{n} 10^k $ is another geometric series:
$$ \sum_{k=1}^{n} 10^k = 10 + 10^2 + \ldots + 10^n = \frac{10(10^n - 1)}{9} $$

The sum $ \sum_{k=1}^{n} 1 = n $.

Substituting these into the expression for $ S_n $:
$$ S_n = \frac{8}{9} \left( \frac{10(10^n - 1)}{9} - n \right) $$

$$ S_n = \frac{8}{81} \left( 10^{n+1} - 10 - 9n \right) $$

This matches Option 3:
$$ \frac{8}{81}[10^{n+1} - 9n - 10] $$

Now, let's explain why the other options are incorrect:

- **Option 1: $\frac{8}{81}[10^{n+1} - 9n + 10]$**: This option incorrectly adds 10 instead of subtracting it, which does not match the derived formula.

- **Option 2: $\frac{8}{81}[10^{n} - 9n - 10]$**: This option uses $ 10^n $ instead of $ 10^{n+1} $, which does not account for the extra factor of 10 needed in the sum.

- **Option 4: $\frac{8}{81}[10^{n-1} + 9n - 10]$**: This option uses $ 10^{n-1} $ and incorrectly adds $ 9n $ instead of subtracting it, which does not match the derived formula.

Therefore, Option 3 is the correct answer, as it accurately represents the sum of the series.