
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
588k+ views
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.
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.
Recently Updated Pages
The height of a solid metal cylinder is 20cm Its r-class-10-maths-ICSE

If a train crossed a pole at a speed of 60kmhr in 30 class 10 physics CBSE

Name the Writs that the High Courts are empowered to class 10 social science CBSE

A tower is 5sqrt 3 meter high Find the angle of el-class-10-maths-CBSE

Immediate cause of variations of A Mutations B Environmental class 10 biology CBSE

A rectangular container whose base is a square of side class 10 maths CBSE

Trending doubts
Why is there a time difference of about 5 hours between class 10 social science CBSE

Why is Sardar Vallabhbhai Patel called the Iron man class 10 social science CBSE

Tropical deciduous trees shed their leaves in the dry class 10 social science CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Write an application to the principal requesting five class 10 english CBSE

