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