The sum to n terms of the series (1)/(3)+(5)/(9)+(19)/(27)+(65)/(81)+…is (A) n-((3^n-2^n))/(2^n)

step 1 Note that one -third can be rewritten as 1 -2 -3rd, and 5 -9s can be rewritten as 1 -4 -9s, and

The sum to n terms of the series...

The sum to $n$ terms of the series $\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\frac{65}{81}+\ldots$ is
(A) $n-\frac{\left(3^{n}-2^{n}\right)}{2^{n}}$
(B) $n-\frac{2\left(3^{n}-2^{n}\right)}{3^{n}}$
(C) $2^{n}-1$
(D) $3^{n}-1$

Key Concepts

Recurrence Relations

Sometimes, the pattern in a series involves differences that lead naturally to a recurrence relation. This is a way of expressing each term as a function of its predecessors. Recognizing and solving such relations can provide a closed-form expression for the terms, which then facilitates the derivation of the sum to n terms.

Series Summation Techniques

Summing a series often requires combining methods that include finding a general term from an observed pattern, and then using summation formulas or manipulations to compute the total. This concept broadly covers techniques like rewriting series, employing transformation methods, or breaking a series into simpler parts whose sums are known.

Geometric Progressions

A key concept in series analysis is the geometric progression, where each term is a constant multiple of the previous term. In the given series, the denominators are powers of a constant (such as 3), representing a typical geometric behavior. Understanding geometric progressions allows one to apply known formulas for the sum of such sequences or incorporate them within more complex series.

Pattern Recognition in Sequences

Identifying the underlying structure in a series is fundamental when its terms are not immediately obvious. This involves analyzing the sequence of numerators or denominators (or both) to discern a regular pattern or recurrence. Recognizing such patterns can help in formulating a general term for the sequence, which is crucial for further analysis and summation.

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