Question

Find the reciprocal of the sum of the first 30 terms of the AP: - 2, - 5, - 8, - 11,………
(a). \[\dfrac{2}{{1365}}\]
(b). \[\dfrac{{ - 1}}{{1365}}\]
(c). \[\dfrac{1}{{1365}}\]
(d). \[\dfrac{{ - 1}}{{1265}}\]

Answer 1

Hint: The sum of n terms of an AP is given by the formula \[{S_n} = \dfrac{n}{2}(2a + (n - 1)d)\]. Find the common ratio and use this formula to find the reciprocal of the sum of 30 terms of the given AP.

Complete step-by-step answer:
An arithmetic progression (AP) is a sequence of numbers such that the difference between the consecutive terms is constant. This constant term is called the common difference.
We are given an AP as follows:
- 2, - 5, - 8, - 11, ………
We need to find the reciprocal of the sum of the 30 terms of this AP.
We first find the common ratio of the AP using the first two terms as follows:
\[d = - 5 - ( - 2)\]
Simplifying we get:
\[d = - 5 + 2\]
\[d = - 3\]
The formula to calculate the sum of n terms of an AP with first term a and common difference d is given as follows:
\[{S_n} = \dfrac{n}{2}(2a + (n - 1)d)\]
Hence, the sum of the first 30 terms of the AP with the first term – 2 and common difference – 3 is given as follows:
\[{S_{30}} = \dfrac{{30}}{2}(2( - 2) + (30 - 1)( - 3))\]
Simplifying, we have:
\[{S_{30}} = 15( - 4 + (29)( - 3))\]
\[{S_{30}} = 15( - 4 - 87)\]
\[{S_{30}} = 15( - 91)\]
\[{S_{30}} = 15( - 91)\]
\[{S_{30}} = - 1365\]
Now, we find the reciprocal of this sum as follows:
\[R = \dfrac{1}{{{S_{30}}}}\]
\[R = - \dfrac{1}{{1365}}\]
Hence, the correct answer is option (b).

Note: If you by mistake calculate the value of the common difference as 3 instead of – 3, you will get the sum as the positive answer and your answer will be wrong. Hence, take care of the negative sign.