Question

5 + 13 + 21 + ⋯ + 181 = ?

Answer 1

Given sequence is 5, 13, 21, …….., 181 

First term of AP is a = 5. 

And common difference of AP is d = a₂ – a₁ = 13 – 5 = 8. 

Let the n^th term of AP is a_n = 181 

∴ a + (n – 1)d = 181 (∵ a_n = a + (n – 1)d) 

⇒ 5 + (n – 1)8 =181 

⇒ 8 (n – 1) = 181 – 5 = 176

⇒ n – 1 = \(\frac{176}{8}=22\)

⇒ n = 22 + 1 = 23.

Hence, 181 is 23^rd term of given AP. 

∵ Sum of first n terms is S_n = \(\frac{n}{2}\) (a + a_n) = \(\frac{23}{2}\) (5 + 181) (∵ n = 23, a = 5 & a_n = 181)

= \(\frac{23}{2}\times186\) = 23 × 93 = 2139.

Hence, 5 + 13 + 21 + ⋯ + = 181 = 2139.