Let there be n numbers in G.P. whose common ratio is r and...

Let there be $n$ numbers in G.P. whose common ratio is $r$ and $S_{m}$ denotes the sum of their first $m$ terms. The sum of their products taken two at a time is $k S_{n} S_{n-1}$ where $k=$
(A) $\frac{r-1}{r}$
(B) $\frac{r-1}{r+1}$
(C) $\frac{r}{r+1}$
(D) None of these

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Key Concepts

Geometric Progression

A geometric progression is a sequence in which each term is derived by multiplying the previous term by a constant factor known as the common ratio. This concept is foundational in algebra and series analysis, allowing for the derivation of formulas for the nth term and various sums associated with the sequence.

Sum of a Geometric Series

The sum of a geometric series is given by a specific formula that depends on the common ratio. When the first term and the common ratio are known, the sum of the first n terms can be expressed in closed form. This formula is crucial in relating individual terms, overall series behavior, and establishing connections between various sums in problems involving geometric sequences.

Symmetric Sums and Pairwise Products

Symmetric sums involve sums over products of elements taken from a sequence in a symmetric, or unordered, way. In particular, the sum of the products taken two at a time requires considering all unique pairs of terms. This concept is important in combinatorial analysis and algebra, as it often links back to other expressions involving the sum of the entire sequence through algebraic manipulation.

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