Find the sum to the series $1. n + 2 (n - 1) + 3 (n - 2) + . . . + n .1$

Open in App

Solution

Given series: $1. n + 2 (n - 1) + 3 (n - 2) + \dots . . + n .1$
$r^{t h}$ term $= r (n - r + 1)$
The given series is summation of $r (n - r + 1)$ where $r$ varies from $1$ to $n$ .
$∴ S u m = n \sum r = 1 [(n + 1) r - r^{2}] = \frac{(n + 1) (n) (n + 1)}{2} - \frac{n (n + 1) (2 n + 1)}{6}$
$S u m = \frac{n (n + 1)}{2} [n + 1 - \frac{2 n + 1}{3}]$
Or, $S u m = \frac{n (n + 1)}{2} [\frac{3 n + 3 - 2 n - 1}{3}]$
$∴ S u m = \frac{n (n + 1)}{2} [\frac{n + 2}{3}] = \frac{n (n + 1) (n + 2)}{6}$ Ans.

Suggest Corrections

Similar questions

1 \times n + 2 (n - 1) + 3 (n - 2) + \dots \dots + (n - 1) \times 2 + n \times 1

n^{t h}

2 n^{3} + 3 n^{2} - 1

2 C_{0} + \frac{C_{1}}{2} \cdot 2^{2} + \frac{C_{2}}{3} \cdot 2^{3} + \frac{C_{3}}{4} \cdot 2^{4} + \dots + \frac{C_{n}}{n + 1} \cdot 2^{n + 1}

(C_{r} =^{n} C_{r})

\frac{1}{(n + 1)} + \frac{1}{2 {(n + 1)}^{2}} + \frac{1}{3 {(n + 1)}^{3}} + \dots

Join BYJU'S Learning Program