Question

The 13th term of an ap is 4 times its third term if its fifth term is 16 then find the first term and common difference

Answer 1

The first term of AP = a
Common difference = d

The 13th term of AP = a + 12d
The 3rd term of AP = a + 2d

According to question :

a + 12d = 4(a + 2d)

a + 12d = 4a + 8d

12d - 8d = 4a - a

4d = 3a

d = 3a/4 ...(1)

The fifth term of AP = a + 4d = 16

a + 4d = 16 ...(2)

put the value of d in equation (2) :

a + 4 × 3a / 4 = 16

a + 3a = 16

4a = 16

a = 16 / 4

a = 4

put the value of a in equation (1) :

d = 3a / 4 = 3 × 4 / 4 = 3

Thus

The first term of AP = a = 4
Common difference = d = 3

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