Sum of $n$ terms of the series $8+88+888+.$ equals
- $\dfrac {8}{18}[10^{n+1}-9n-10]$
- $\dfrac {8}{18}[10^{n}-9n-10]$
- $\dfrac {8}{18}[10^{n+1}+9n-10]$
- $None\ of\ these$
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A
Correct answer
AI explanation
Rewrite each term by factoring out 8, changing the series into 8/9 multiplied by (9 + 98 + 998 + ... up to n terms). Express each number inside the parentheses as (10 - 1), (100 - 2), (1000 - 3), up to (10^n - n), separating the series into the geometric progression 10 + 100 + 1000 + ... + 10^n and the arithmetic progression 1 + 2 + 3 + ... + n. The geometric progression sum is 10(10^n - 1)/9 and the arithmetic progression sum is n(n+1)/2. Combining these gives 8/9 * [10(10^n - 1)/9 - n(n+1)/2], which algebraically simplifies to the equivalent form 8/18 * [10^(n+1) - 9n - 10].