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Sequence and Series

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Question : 82 of 86

Marks: +1, -0

The Sum of First

n

terms of an A.P is

5 n^{2}+7 n

10^{\;\text{th }\;}

term is

101
96
84
102

Solution:

Concept:

For any arithmetic progression, the

n

th term

a_n

can be found using

a_n = S_n - S_{n-1}

.

Explanation:

Given

S_n = 5n^2 + 7n

.

First, find

S_{10}

and

S_9

:

S_{10} = 5(10)^2 + 7(10) = 5 \cdot 100 + 70 = 500 + 70 = 570

.

S_9 = 5(9)^2 + 7(9) = 5 \cdot 81 + 63 = 405 + 63 = 468

.

Thus,

10

th term

a_{10} = S_{10} - S_9 = 570 - 468 = 102

.

Alternatively, derive general term:

a_n = S_n - S_{n-1} = (5n^2+7n) - [5(n-1)^2 + 7(n-1)] = 10n + 2

.

For

n=10

,

a_{10} = 10(10) + 2 = 102

.

Answer:

The

10^\text{th}

term is

102

, which corresponds to option D.

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