First slide

Arithmetic progression

Question

The sum of n terms of m A.P.s are S₁, S₂, S₃, …, S_m. If the first term and common difference are 1, 2, 3, …, m respectively, then S₁ + S₂ + S₃ + … + $S_{m}$ =

Moderate

A

$\frac{1}{4} mn (m + 1) (n + 1)$
B

$\frac{1}{2} mn (m + 1) (n + 1)$
C

$mn (m + 1) (n + 1)$
D

None of these

Solution

We have, $\begin{array}{l} S_{1} = (n / 2) [2 \cdot 1 + (n - 1) \cdot 1] \\ S_{2} = (n / 2) [2 \cdot 2 + (n - 1) \cdot 2] \\ S_{m} = (n / 2) [2 \cdot m + (n - 1) \cdot m] \end{array}$
$\begin{array}{l} ∴ S_{1} + S_{2} + \dots + S_{m} \\ = n (1 + 2 + 3 \dots + m) + \frac{n (n - 1)}{2} \times (1 + 2 + \dots + m) \\ = \frac{m (m + 1)}{2} (n + \frac{n^{2} - n}{2}) \\ = \frac{m (m + 1)}{2} \cdot \frac{n (n + 1)}{2} = \frac{1}{4} mn (m + 1) (n + 1) \end{array}$

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