Question

For what value of n, the nth terms of the two A.P.s: 63, 65, 67, ... and 3, 10, 17, ... are equal ?

Answer 1

Hint: Use the formula for the nth term of an A.P.: ${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$. From the first term, second term and the common difference of both the A.Ps. Using these, find the nth terms of the A.P.s. equate these to find the value of n which is our final answer.

Complete step-by-step answer:
In this question, we are given two A.P.s: 63, 65, 67, ... and 3, 10, 17, …

We need to find the value of n such that the nth terms of the two A.P.s are equal.

For this question, we will use the formula for nth term of an A.P.:

${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$

First A.P is 63, 65, 67, …

First term of this A.P, ${{a}_{1}}$ ​= 63

Second term of this A.P, ${{a}_{2}}$​ = 65

Common difference, d = ${{a}_{2}}$ ​− ${{a}_{1}}$​ = 65 – 63 = 2

Hence, the nth term of this A.P is given by ${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$

\[~{{a}_{n}}=63\text{ }+\text{ }2\left( n\text{ }-1 \right)\]

\[~{{a}_{n}}=63\text{ }+2n-2\]

\[~{{a}_{n}}=61+2n\] …(1)

Similarly, we will find the nth term of the second A.P.

Second A.P. is 3, 10, 17, …

First term of this A.P, ${{a}_{1}}$ ​= 3

Second term of this A.P, ${{a}_{2}}$​ = 10

Common difference, d = ${{a}_{2}}$ ​− ${{a}_{1}}$​ = 10 – 3 = 7

Hence, the nth term of this A.P is given by ${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$

\[~{{a}_{n}}=3\text{ }+\text{ 7}\left( n\text{ }-1 \right)\]

\[~{{a}_{n}}=3+7n-7\]

\[~{{a}_{n}}=-4+7n\] …(2)

We have to find that term (n) for which the nth term of both A.P are equal

So, we will equate equation (1) and equation (2).

61 + 2n = -4 + 7n

65 = 5n

n = 13

Hence, the 13th term of both the A.P.s are equal.

Note: In this question, it is very important to know about the general formula for the nth term of an A.P. The formula for the nth term of an A.P. is given by: ${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$.