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Question

An AP has the first term as 2 and the common difference as 4. Which term in the AP is the first multiple of 23 to appear in the AP?

A
11th term
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B
12th term
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C
14th term
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D
35th term
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Solution

The correct option is B 12th term
Given a = 2 and d = 4
nth term an=a+(n1)d
Any multiple of 23 can be taken as 23k (k = 1, 2, 3, ...)
Given, an=23k
a+(n1)d=23k
2+(n1)4=23k
(n1)4=23k2
n1=23k24
Since n must be a natural number, as n is the number of terms of an AP, the expression 23k24 must be a whole number.
When we put k = 1, 23k24=23×124=214 which is not a whole number
When we put k = 2, 23k24=23×224=444=11 which is a whole number
Hence, 23×2=46 is the first multiple of 23 to appear in the AP
Now, since we have used the formula for finding the nth term, the value of n should give us which term of the AP is 46.
We know that n -1 = 11, hence n = 12
So, 12th term in the AP is the first multiple of 23 to appear in the AP.

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