Find the sum of first 20 terms of an A.P. whose nth term is given by $a_n = 5 + 2n$. Can 52 be a term of this AP.?

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Find the sum of first 20 terms of an A.P. whose nth term is given by $a_n = 5 + 2n$. Can 52 be a term of this AP.?

Posted by: Mohit Bhardwaj | Created at: Mar 13, 2025 | Last updated: Mar 13, 2025

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Rameshwar · Accepted Answer · 2025-03-13T13:03:43+00:00

Given the nth term of an arithmetic progression:
$ a_n = 5 + 2n $

To find the sum of the first 20 terms, we use the sum formula:
$ S_n = \frac{n}{2} (2a + (n-1)d) $

Here,
First term $ a = a_1 = 5 + 2(1) = 7 $,
Common difference $ d = a_2 - a_1 $

$= (5 + 2(2)) - (5 + 2(1)) $

$= 9 - 7 = 2 $,
Number of terms $ n = 20 $.

Substituting in the formula:
$ S_{20} = \frac{20}{2} (2(7) + (20-1)(2)) $
$ = 10 (14 + 38) $
$ = 10 \times 52 $
$ = 520 $

Thus, the sum of the first 20 terms is $ 520 $.

Now, checking if 52 is a term of the sequence:

If 52 is a term, then for some $ n $:
$ 5 + 2n = 52 $

Solving for $ n $:
$ 2n = 47 $
$ n = 23.5 $

Since $ n $ is not a whole number, 52 is not a term of this A.P.