The sum of first 9 terms of an A.P. is 162. The ratio of its 6^th term to its 13^th term is 1:2. Find the first and 15^th term of the A.P.

Let a be the first term and d be the common difference

Now, a₆ / a₁₃ =

=

2a + 10d = a + 12d

a = 2d (i)

Now sum of first n terms of A.P

S_n = [2a + (n – 1) d]

S₉ = [2a + 8d]

162 = 9 (a + 4d)

a + 4d = 18

2d + 4d = 18 [Using (i)]

6d = 18

d =3

Now from (i), we get

a = 2 * 3 = 6

So, first term, a₁ = a = 6

15^th term, a₁₅ = a + 14d = 6 + 14(3)

= 6 + 42 = 48

Therefore, First term is 6 and 15^th term is 48