Question

The 19th term of an AP is equal to three times its sixth term. If its 9th term is 19,find the AP.

Answer:

The AP is 3, 5, 7, 9, 11, ..............

We are given that the 19th term of the AP is 3 times its 6th term, i.e.,
a + (19-1)d = 3[a + (6-1)d]
⇒a + 18d = 3(a + 5d)
⇒a + 18d = 3a + 15d
⇒a + 8d = 3a + 5d
Now that we are given the 9th term to be 19, a + (9-1)d = 19 which implies that a + 8d = 19 ... (1)
Therefore,
19 = 3a + 5d
Now, (1) also implies that a = 19-8d. Substituting value of a in the above equation, we get:
19 = 3(19-8d) + 5d
⇒19 = 57 - 24d + 5d
⇒19 = 57 - 19d
⇒19d = 38
⇒d = 2

Going back to (1),
a + 8(2) = 19
⇒a + 16 = 19
⇒a = 3


Please feel free to comment below for a detailed discussion/worked out solution to this problem.