Find the sum :

7 + $10\dfrac{1}{2}$ + 14 +……+ 84.

Series :

7 + 10.5 + 14 + ……. + 84

The above series is an A.P. with common difference = 3.5

Let 84 be nth term in the A.P.

We know that,

∴ a_n = a + (n - 1)d

⇒ 84 = 7 + (n - 1) × 3.5

⇒ 84 - 7 = (n - 1) × 3.5

⇒ 77 = (n - 1) × 3.5

⇒ $\dfrac{77}{3.5}$ = (n - 1)

⇒ 22 = (n - 1)

⇒ n = 22 + 1

⇒ n = 23.

We know that,

Sum of n terms of an A.P. is given by,

∴ S_n = $\dfrac{n}{2}$ (a + l)

⇒ S₂₃ = $\dfrac{23}{2}$ (7 + 84)

= $\dfrac{23}{2}$ (91)

= $\dfrac{2093}{2} = 1046\dfrac{1}{2}$

Hence, S₂₃ = $1046\dfrac{1}{2}$.