If an A.P. consists of $n$ terms with first term $a$ and $n$th term $l$ show that the sum of the $m$th term from the beginning and the $m$th term from the end is $(a + l)$

Given:

An A.P. consists of $n$ terms with first term $a$ and $n$th term $l$.

To do:

We have to show that the sum of the $m$th term from the beginning and the $m$th term from the end is $(a + l)$.

Solution:

First term $a_1=a$

nth term $a_n=l$

Let the common difference of the A.P. be $d$.

We know that,

$a_n=a+(n-1)d$

Therefore,

$m$th term from the beginning $a_m=a+(m-1)d$

$m$th term from the end $=l-(m-1)d$

Sum of the $m$th term from the beginning and the $m$th term from the end $=a+(m-1)d+l-(m-1)d$

$=a+l$

Hence proved.

Related Articles If $(m + 1)$th term of an A.P. is twice the $(n + 1)$th term, prove that $(3m + 1)$th term is twice the $(m + n + 1)$th term.
If $m$ times the $m^{th}$ term of an AP is equal to $n$ times its $n^{th}$ term. find the $( m+n)^{th}$ term of the AP.
The $m^{th}$ term of an arithmetic progression is $x$ and the $n^{th}$ term is $y$. Then find the sum of the first $( m+n)$ terms.
If $m^{th}$ term of an $A.P.$ is $\frac{1}{n}$ and $n^{th}$ term of another $A.P.$ is $\frac{1}{m}$. Then, show that $(mn)^{th}$ term is equal to $1$.
For an AP, if \( m \) times the mth term equals \( \mathrm{n} \) times the \( n \) th term, prove that \( (m+n) \) th term of the AP is zero. \( (m ≠ n) \).
The $14^{th}$ term of an A.P. is twice its $8^{th}$ term. If its $6^{th}$ term is $-8$, then find the sum of its first $20$ terms.
The sum of first $m$ terms of an A.P. is $4m^2 – m$. If its $n$th term is 107, find the value of $n$. Also, find the 21st term of this A.P.
The $4^{th}$ term of an A.P. is zero. Prove that the $25^{th}$ term of the A.P. is three times its $11^{th}$ term.
If the 10th term of an A.P. is 21 and the sum of its first ten terms is 120, find its $n$th term.
The sum of first $n$ terms of an A.P. is $5n^2 + 3n$. If its $m$th term is 168, find the value of $m$. Also, find the 20th term of this A.P.
Find first term, common difference and $5^{th}$ term of the sequence which have the following $n^{th}$ term: $3n+7$.
If the $2^{nd}$ term of an A.P. is $13$ and the $5^{th}$ term is $25$, what is its $7^{th}$ term?
If the $n^{th}$ term of an AP is $\frac{3+n}{4}$, then find its $8^{th}$ term.
Find the $5^{th}$ term of an A.P. of $n$ terms whose sum is $n^2−2n$.
If the sum of the first $n$ terms of an AP is $\frac{1}{2}(3n^2+7n)$ then find its $n^{th}$ term. Hence, find its $20^{th}$ term.
Kickstart Your Career
Get certified by completing the course

Get Started