Question

The eighth term of an AP is half its second term and the eleventh term exceeds one-third of its fourth term by 1. Find the 15^th term.

Answer 1

Let a and d the first term and common difference of an AP, respectively

Now, by given condition, `a_8 = 1/2 a_2`

⇒ `a + 7d = 1/2 (a + d)`  ......`[because a_n = a + (n - 1)s]`

⇒ `2a + 14d = a + d`

⇒ `a + 13d` = 0  ......(i)

And `a_11 = 1/3 a_4 + 1` ......[Given]

⇒ `a + 10d = 1/3 [a - 3d] + 1`

⇒ `3a + 30d = a + 3d + 3`

⇒ `2a + 27d` = 3

From equations (i) and (ii)

2(– 13d) + 27d = 3

⇒ – 26d + 27d = 3

⇒ d = 3

From equation (i),

`a + 13(3)` = 0

⇒ a = – 39

∴ a₁₅ = a + 14d

= – 39 + 14(3)

= – 39 + 42

= 3