If in an A.P., Sn=p, n^2 and Sm=p, m^2 where Sp denotes the...

If in an A.P., $S_{n}=p, n^{2}$ and $S_{m}=p, m^{2}$ where $S_{p}$ denotes the sum of $r$ terms of the A.P., then $S_{p}$ is equal to
(A) $\frac{1}{2} p^{3}$
(B) $m m p$
(C) $p^{3}$
(D) $(m+n) p^{2}$

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Key Concepts

Algebraic Manipulation

Techniques such as substitution, rearrangement, and factorization are used to simplify and solve the equations derived from the properties of an arithmetic progression. Mastery of these operations is important for deriving general expressions and solving for specific quantities in sequence problems.

Simultaneous Equations

In problems where multiple expressions are given for the sum of terms (for different numbers of terms), simultaneous equations are employed to solve for unknown variables like the first term and the common difference. This method is fundamental in establishing relationships between different given sums.

Arithmetic Progression

An arithmetic progression is a sequence in which each term after the first is obtained by adding a constant difference to the preceding term. This concept is central to problems involving series because it establishes a predictable linear relationship between terms.

Sum Formula for Arithmetic Progression

The sum of the first n terms of an arithmetic progression can be computed using the formula S? = n/2 [2a + (n - 1)d], where a represents the first term and d the common difference. This formula is key to linking individual term properties with cumulative values in the series.

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