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The sum of the first 7 terms of an AP is 182. If its 4th and 17th terms are in the ratio 1:5, then the AP is.

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2, 10, 18, 26,...

1, 7, 13, 19,...

1, 3, 5, 7,...

2, 4, 6, 8,...

answer is C.

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Detailed Solution

It is given that the sum of the first 7 terms is 182.

S 7 = 182

Sum of the n terms is given by,

S n = n 2 [2 a + (n − 1) d]

Substitute the known values in the formula,

72 (2 a + 6 d) = 182

a + 3 d = 26

……(1)
Now, its 4th and 17th terms are in the ratio 1:5.

a 4 : a 17 = 1 : 5

……(2)
The nth term of an AP is,

a n = a + (n − 1) d

So,

a 4 = a + (4 − 1) d = a + 3 d ...... (3)

And,

a 17 = a + (17 − 1) d = a + 16 d ....... (4)

Substitute the value of (3), (4) in (2),

a + 3 d a + 16 d = 15

By cross multiplication and by simplifying,

5 a + 15 d = a + 16 d 5 a − a = 16 d − 15 d

By simplification,

d = 4 a

. ……(2)
Substitute the value of d in equation (1),

a + 3 × 4 a = 26 13 a = 26 a = 2613 a = 2

Find d by substituting a = 2 in (2),
d = 8
Generally AP is in the form of

a, a + d, a + 2 d, ...

So, by substituting the appropriate values, we get the AP as follows:
2, 10, 18, 26....
Hence, AP is 2, 10, 18, 26....
Therefore, option 3 is correct.

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