164, 160, 156, 152, ….. are in Arithmetic Progression (A.P.). Find :

(a) which term is equal to 0.

(b) the sum of its first 20 terms.

Given,

First term (a) = 164

Common difference (d) = 160 - 164 = -4

(a) Let n^th term be zero.

⇒ a_n = 0

⇒ a + (n - 1)d = 0

⇒ 164 + (n - 1)(-4) = 0

⇒ 164 - 4n + 4 = 0

⇒ 168 - 4n = 0

⇒ 4n = 168

⇒ n = $\dfrac{168}{4}$

⇒ n = 42.

Hence, 42nd term is equal to 0.

(b) By formula,

$S_n = \dfrac{n}{2}\Big[2a + (n - 1)d\Big]$

Substituting values we get :

$S${20} = \dfrac{20}{2}\Big[2 \times (164) + (20 - 1) \times (-4)\Big] \\[1em] \Rightarrow S{20} = 10[328 + (19) \times (-4)] \\[1em] \Rightarrow S{20} = 10[328 - 76] \\[1em] \Rightarrow S{20} = 10 \times 252 \\[1em] \Rightarrow S_{20} = 2520.

Hence, sum of first 20 terms = 2520.