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If the ratio of $n^{th}$ terms of two A.P.'s is $(2 n + 8) : (5 n - 3)$ , then find the ratio of the sums of their $n$ terms.

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(2 n + 18) : (5 n + 3)

(5 n - 18) : (2 n + 18)

(2 n + 18) : (5 n - 1)

(3 n + 18) : (4 n + 1)

answer is C.

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Detailed Solution

Given, the ratio of

n^{th}

terms of A.P is

(2 n + 8) : (5 n - 3)

.
It can be written as

Ratio = \frac{2 n + 8}{5 n - 3}

Add and subtract 2 in numerator and 5 in denominator,

Ratio = \frac{2 n - 2 + 2 + 8}{5 n - 5 + 5 - 3}

Ratio = \frac{10 + (n - 1) \times 2}{2 + (n - 1) \times 5}

The

n^{th}

term of an AP is calculated using

a_{n} = a + (n - 1) d

A.P. in numerator,

a_{1} = 10

and

d_{1} = 2

For the denominator,

a_{2} = 2

and

d_{2} = 5

Sum of n terms

S_{n} = \frac{n}{2} [2 a + (n - 1) d]

Sum of n terms of the A.P. in numerator is,

S_{n} = \frac{n}{2} [2 \times 10 + (n - 1) \times 2]

Sum of n terms of the A.P. in denominator is,

S_{n} = \frac{n}{2} [2 \times 2 + (n - 1) \times 5]

Then, the ratio of the sums be,

Ratio = \frac{\frac{n}{2} [20 + 2 (n - 1)]}{\frac{n}{2} [4 + 5 (n - 1)]}

Ratio = \frac{2 n + 18}{5 n - 1}

Ratio = (2 n + 18) : (5 n - 1)

Hence, option 3 is correct.

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