The 5th and 15th terms of an A.P. are 13 and - 17 respectively. Find the sum of first 21 terms of the A.P.

Given: a₅ = 13 and a₁₅ = -17 and n = 21

We know that,

a₅ = a + 4d = 13 …(i)

and a₁₅ = a + 14d = -17 …(ii)

Solving the linear equations (i) and (ii), we get

a + 4d – a – 14d = 13 – (-17)

⇒ -10d = 13 + 17

⇒ -10d = 30

⇒ d = -3

Putting the value of d in eq. (i), we get

a + 4(-3) = 13

⇒ a = 13 + 12 = 25

Now, we have to find the sum of first 21 terms.

⇒ S₂₁ = 21 [25 + 10 × (-3)]

⇒ S₂₁ = 21 [-5]

⇒ S₂₁ =-105