Question

The 19^th term of an AP is equal to 3 times its 6^th term. If its 9^th term is 19, find the AP.

Answer 1

Let a be the first term and d be the common difference of the AP. Then,

a₁₉ = 3a₆   (Given)

\(\Rightarrow\) a + 18d = 3(a + 5d)   [a_n = a + (n - 1)d]

\(\Rightarrow\) a + 18d = 3a + 15d

\(\Rightarrow\) 3a - a = 18d - 15d

\(\Rightarrow\) 2a = 3d   .......(1)

Also

a₉ = 19   (Given)

\(\Rightarrow\) a + 18d = 19   ....(2)

From (1) and (2), we get

\(\frac{3d}{2}+8d=19\)

\(\Rightarrow\) \(\frac{3d\,+\,16d}{2}=19\)

\(\Rightarrow\) 19d = 38

\(\Rightarrow\) d = 2

Putting d = 2 in (1), we get

2a = 3 x 2 = 6

\(\Rightarrow\) a = 3

So,

Hence, the AP is 3,5,7,9,........