Multiple choice

The sum of the series, $\displaystyle \frac{1}{2.3}\cdot 2+ \frac{2}{3.4}\cdot 2^2+\frac{3}{4.5}\cdot 2^3+ ......$ to n terms is _____.

  1. $\dfrac{2^{n+1}}{n+2}+1$
  2. $\dfrac{2^{n+1}}{n+2}-1$
  3. $\dfrac{2^{n+1}}{n+2}+2$
  4. $\dfrac{2^{n+1}}{n+2}-2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The n-th term is n/((n+1)(n+2)) * 2^n. Using partial fractions, n/((n+1)(n+2)) = 2/(n+2) - 1/(n+1). Summing these terms shows the series telescopes to 2^(n+1)/(n+2) - 1.

AI explanation

The nth term of the series is given by n times 2 to the power of n divided by the quantity n plus 1 multiplied by n plus 2. We split the term into partial fractions to get 2 raised to n times the quantity 1 divided by n plus 1, minus 1 divided by n plus 2. Summing this from 1 to n produces telescoping terms, leaving the final sum as 4 minus 2 divided by n plus 2. This simplifies to the fraction 2 raised to n plus 1 divided by n plus 2, minus 1.

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