Question

1. The nth terms of the two series 3 + 10 + 17 + .... and 63 + 65 + 67 + .... are equal, then the value of ‘n’
is [ ]
a) 9 b) 13 c) 19 d) 11

Answer 1

Given:

• Two series

⟼ 3 + 10 + 17 + ....

⟼ 63 + 65 + 67 + ....

• nth terms of both the given series are equal

To Find:

• Value of n

Solution:

We know that,

• A series is said to be in A.P. if the difference of each adjacent term term are same i.e. they are having common difference.
• nth term of an A.P. is expressed as

⟼   $\bf{a_{n}=a+(n-1)d}$

where,

a is the first term of A.P.

n is no. of term of A.P.

d is common difference of A.P.

Now,

Since, both the series are having common difference throughout the A.P.

So, both are in A.P.

In first A.P.,

First term,a =3

Common difference,d = 7

In second A.P.,

First term,a' = 63

Common difference,d' = 2

Let the nth term of first and second A.P. be $a_{n}$ and $a'_{n}$ respectively

Now, it is given that nth term of both the A.P.'s are equal, so

$\sf{nth\:term\:of\:First\:A.P.=nth\:term\:of\:Second\:A.P.}$

$\longrightarrow\sf{a_{n}=a'_{n}}$

$\longrightarrow\sf{a+(n-1)d=a'+(n-1)d'}$

$\longrightarrow\sf{3+(n-1)7=63+(n-1)2}$

$\longrightarrow\sf{3+7n-7=63+2n-2}$

$\longrightarrow\sf{7n-2n=63+7-3-2}$

$\longrightarrow\sf{5n=65}$

$\longrightarrow\sf{n=\cancel{\dfrac{65}{5}}}$

$\longrightarrow\sf\pink{n=13}$

Hence, the correct answer is b) 13.