Name: nᵗʰ Term From the End of an AP
Uploaded: 2019-07-19
Duration: 2 min 48 s
Description: Let's say you are given a number of terms in an arithmetic progression, and you have to find a certain nth term from the end. To learn how to find it, watch...

Question

In an AP, if the sum of the first

m

terms is equal to

n

and that of the first

n

terms is equal to

m

, then prove that the sum of the first

(m + n)

terms is

- (m + n)

.

Accepted Answer

Let $a$ be the first term and $d$ be the common difference of the given AP.

Then, $S_{m} = \frac{m}{2} [2 a + (m - 1) d]$

$\Rightarrow n = \frac{1}{2} [2 am + (m^{2} - m) d]$

$\Rightarrow 2 n = 2 am + (m^{2} - m) d . . . . . (i)$

and $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

$\Rightarrow m = \frac{1}{2} [2 an + (n^{2} - n) d]$

$\Rightarrow 2 m = 2 an + (n^{2} - n) d . . . . . (ii)$

On subtracting eq $(ii)$ from eq $(i)$ , we get

$2 n - 2 m = [2 am + (m^{2} - m) d] - [2 an + (n^{2} - n) d]$

$\Rightarrow 2 (n - m) = 2 am + (m^{2} - m) d - 2 an - (n^{2} - n) d$

$\Rightarrow 2 (n - m) = 2 am + m^{2} d - md - 2 an - n^{2} d + nd$

$\Rightarrow 2 (n - m) = 2 am - 2 an + m^{2} d - n^{2} d - md + nd$

$\Rightarrow - 2 (m - n) = 2 a (m - n) + (m^{2} - n^{2}) d - (m - n) d$

$\Rightarrow - 2 (m - n) = (m - n) [2 a + (m + n) d - d]$

$\Rightarrow - 2 = [2 a + (m + n - 1) d] . . . . . (iii)$

Sum of $(m + n)$ terms of the given AP

$S_{m + n} = \frac{m + n}{2} [2 a + (m + n - 1) d]$

$\Rightarrow S_{m + n} = \frac{m + n}{2} \times - 2$ (from eq $(iii)$

$\Rightarrow S_{m + n} = - (m + n)$ . Hence proved.

In an AP, if the sum of the first $m$ terms is equal to $n$ and that of the first $n$ terms is equal to $m$ , then prove that the sum of the first $(m + n)$ terms is $- (m + n)$ .

Important Questions on Arithmetic Progression (AP)

Learn and Prepare for any exam you want !

In an AP, if the sum of the first m terms is equal to n and that of the first n terms is equal to m, then prove that the sum of the first m+n terms is -(m+n).

Important Questions on Arithmetic Progression (AP)

Learn and Prepare for any exam you want !

In an AP, if the sum of the first $m$ terms is equal to $n$ and that of the first $n$ terms is equal to $m$ , then prove that the sum of the first $(m + n)$ terms is $- (m + n)$ .