Question

If mth terms of the series $63+65+67+69+\ldots .$ and $3+10+$ $17+24+\ldots$ be equal , then $m=$
(1) 11
(2) 12
(3) 13
(4) 15

Answer 1

Hint: In an arithmetic progression (AP) sequence of numbers, the difference between any two consecutive integers is always the same. It is also known as an arithmetic sequence.

Formula Used:
The mth term $=a+(m-1) d$

Complete step by step Solution:
For the series, $63+65+67+69+\ldots .$
First term, $a=63$
In arithmetic progression, d is used to denote a common difference. The difference between the term that comes after the one that came before. In a nutshell, we can state that a sequence is in A.P. if the common difference between successive elements is constant.
Common difference, $d=2$
mth term $=a+(m-1) d$
$=63+(m-1) 2$
$=63+2 m-2$
$=61+2 m \ldots(i)$
For the series $3+10+17+24+\ldots .$
First term, $a=3$
Common difference, $d=7$
To find any term of the AP (Arithmetic Progression), use the mth term. A term for AP is often created by adding its common difference to its prior term. However, we can locate any AP term using the nth term of the AP formula without knowing its prior term.
$m$ th term $=a+(m-1) d$
Substituting we get
$=3+(\mathrm{m}-1) 7$
$=3+7 \mathrm{~m}-7$
$=-4+7 \mathrm{~m} \ldots \text {.(ii) }$
Equating (i) and (ii)
$61+2 m=-4+7 m$
Simplify the expression
$65=5 m$
$m=65 / 5$
$=13$

Hence, the correct option is 3.

Note: The algebraic sequence of numbers known as arithmetic progression is one in which every subsequent term's difference is the same. It can be attained by multiplying each preceding phrase by a predetermined number.
A good example of an arithmetic progression (AP) is the series 2,6,10,14,..., which follows a pattern in which each number is created by adding 4 to the previous term. The nth term in this series equals $4n-2$. You may get the terms of the series by changing the nth term to $n=1,2,3,..$.