Question

The 13^th terms of an AP is 4 times its 3^rd term. If its 5^th term is 16, Find the sum of its first 10 terms.  

Answer 1

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

a₁₃ =  4 × a₃                      (Given)

⇒ a  +12d  = 4 (a + 2d)                 [ a_n  = a + (n-1) d]

⇒  a +12d = 4a +8d

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

Also,

a₅ = 16              (Given) 

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

Solving (1) and (2), we get

a +3a = 16

 ⇒ 4a = 16 

⇒  a =4

Putting a = 4 in (1), we get

4d = 3× 4 = 12 

⇒  d = 3

`"Using the formula ", S_4 = n/2 [ 2a + (n-1) d ] `, we get

`S_10 = 10/2 [2xx4+(10-1) xx3]`

=5 ×  ( 8 + 27)

= 5× 35

=175

Hence, the required sum is 175.