The sum of n terms of m A.P.s are S1, S2, S3, …, Sm, If the first term and common difference are

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The sum of n terms of m A.P.s are S1, S2, S3, …, Sm, If the...

The sum of $n$ terms of $m$ A.P.s are $S_{1}, S_{2}, S_{3}, \ldots, S_{m}$, If the first term and common difference are $1,2,3, \ldots, m$ respectively, then $S_{1}+S_{2}+S_{3}+\ldots+S_{m}=$ (A) $\frac{1}{4} m n(m+1)(n+1)$ (B) $\frac{1}{2} m n(m+1)(n+1)$ (C) $m n(m+1)(n+1)$ (D) None of these

The sum of $n$ terms of $m$ A.P.s are $S_{1}, S_{2}, S_{3}, \ldots, S_{m}$, If the first term and common difference are $1,2,3, \ldots, m$ respectively, then $S_{1}+S_{2}+S_{3}+\ldots+S_{m}=$
(A) $\frac{1}{4} m n(m+1)(n+1)$
(B) $\frac{1}{2} m n(m+1)(n+1)$
(C) $m n(m+1)(n+1)$
(D) None of these

Key Concepts

Parameterization in Sequences

Parameterization in sequences refers to expressing the sequence's first term and common difference as functions of an index or parameter. This approach allows the generalization and analysis of a family of sequences simultaneously by substituting the index into the formula for each sequence, which is particularly useful in problems involving multiple related arithmetic progressions.

Summation Over Multiple Sequences

In problems where multiple arithmetic progressions are involved, each with parameters depending on an index (such as different first terms and common differences), it is necessary to apply the sum formula to each progression and then sum the results. This requires techniques in handling combined or double summations, where the summation over one parameter is nested within a summation over another.

Sum Formula for an Arithmetic Series

The sum of the first n terms of an arithmetic progression is given by the formula S = n/2 [2a + (n - 1)d], where a represents the first term and d the common difference. This formula is fundamental for evaluating the total of a sequence and is widely used in problems involving sums of arithmetic progressions.

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers where the difference between any two successive terms is constant. This constant difference, along with the first term, determines all subsequent terms in the sequence, making it essential in various areas of mathematics including series and summation problems.

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