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The ratio of the 11th term to the 18th term of an AP is 2 : 3. The ratio of the $5 t h$ term to the $21 s t$ term is

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1:1

1:2

1:3

1:4

answer is C.

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Detailed Solution

Given, the ratio of the 11th term to the 18th term of an AP is 2:3.
The nth terms of an AP is,

a n = a + n − 1 d

where, a is the first term, d is a common difference and n is the number of terms to be added.
As,

a 11 a 18 = 23

a + 11 − 1 d a + 18 − 1 d = 23 a + 10 d a + 17 d = 23 3 a + 30 d = 2 a + 34 d a = 4 d

.
So, according to the question,
The ratio of the 5th term to the 21st term will be,

a 5 a 21 = a + 5 − 1 d a + 21 − 1 d a 5 a 21 = a + 4 d a + 20 d

.
Substitute [a = 4d] in the expression

a + 4 d a + 20 d

a 5 a 21 = 4 d + 4 d 4 d + 20 d a 5 a 21 = 8 d 24 d a 5 a 21 = 13

.
Thus, the resultant ratio is 1:3.
Therefore, the correct option is 3.

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