The sum of the series $12 + 104 + 1008 + 10016 + ...$ upto $n$ terms is
- $\displaystyle \frac{(10^{n}-1)}{9}+2^{n}$
- $\displaystyle \frac{10(10^{n}-1)}{9}+2(2^{n}-1)$
- $\displaystyle \frac{10^{n}-1}{9}-1$
- $\displaystyle \frac{10(10^{n}-1)}{9}$
Reveal answer
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B
Correct answer
Explanation
The series is (10+2) + (100+4) + (1000+8) + ... = (10^1 + 10^2 + ... + 10^n) + (2^1 + 2^2 + ... + 2^n). The first part is a GP sum: 10(10^n - 1) / 9. The second part is a GP sum: 2(2^n - 1) / (2-1) = 2(2^n - 1).
AI explanation
The terms of the series can be split into the sum of two separate patterns: 10 + 100 + 1000 + ... and 2 + 4 + 16 + .... The first part is a geometric progression with the first term 10 and common ratio 10, yielding a sum of 10(10^n - 1) / 9. The second part is a geometric progression with the first term 2 and common ratio 2, giving a sum of 2(2^n - 1). Adding these two sums together results in 10(10^n - 1) / 9 + 2(2^n - 1).