Question

If the common difference of an A.P is 5, then what is ${{a}_{18}}-{{a}_{13}}$ ?
(A) 5
(B) 20
(C) 25
(D) 30

Answer 1

Hint: For solving questions of this type we should determine the ${{n}^{th}}$ term of the progression series is given by ${{a}_{n}}=a+\left( n-1 \right)d$ where $a$ is the first term and $d$ is the common difference between any two consecutive terms of the progression. And we should evaluate the value of ${{18}^{th}}$ term and ${{13}^{th}}$ term and find the difference.

Complete step-by-step answer:
As given in the question that the common difference of an A.P is 5. This can be simply given as $d=5$.
From the basic concept of progressions we know that the ${{n}^{th}}$ term of the arithmetic progression series is given by ${{a}_{n}}=a+\left( n-1 \right)d$ where $a$ is the first term and $d$ is the common difference between any two consecutive terms of the progression.
Let $a$ be the first term of the given progression.
So here we can find the ${{18}^{th}}$ term by substituting $d=5$
$ a+\left( 18-1 \right)d $
$ = a+17d $
$ = a+17\left( 5 \right) $
$ = a+85 $
And similarly we can find the ${{13}^{th}}$ term by substituting $d=5$
$ \ a+\left( 13-1 \right)d $
$ = a+12d $
$ = a+12\left( 5 \right) $
$ = a+60 $
So we have the value of ${{a}_{18}}=a+85$ and ${{a}_{13}}=a+60$
 The value of ${{a}_{18}}-{{a}_{13}}$ is equal to
$ \left( a+85 \right)-\left( a+60 \right) $
$ = 25 $
So we can conclude that ${{a}_{18}}-{{a}_{13}}=25$.

So, the correct answer is “Option C”.

Note: While solving questions of this type we should be aware of the basic concept of progressions let us discuss about the ${{n}^{th}}$ term for the geometric progression series is given by ${{a}_{n}}=a{{r}^{n-1}}$ where $a$ is the first term and $r$ is the ratio of two consecutive terms. We can also assume the value of the first term as 1 it will make no difference in the answer we get.