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.
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Given,
First term (a) = 164
Common difference (d) = 160 - 164 = -4
(a) Let nth term be zero.
⇒ an = 0
⇒ a + (n - 1)d = 0
⇒ 164 + (n - 1)(-4) = 0
⇒ 164 - 4n + 4 = 0
⇒ 168 - 4n = 0
⇒ 4n = 168
⇒ n = $\frac{168}{4}$
⇒ n = 42.
Hence, 42nd term is equal to 0.
(b) By formula,
$S_n = \frac{n}{2}\Big[2a + (n - 1)d\Big]$
Substituting values we get :
$$\begin{aligned}S_{20} = \frac{20}{2}\Big[2 \times (164) + (20 - 1) \times (-4)\Big] \\[1em] \Rightarrow S_{20} = 10[328 + (19) \times (-4)] \\[1em] \Rightarrow S_{20} = 10[328 - 76] \\[1em] \Rightarrow S_{20} = 10 \times 252 \\[1em] \Rightarrow S_{20} = 2520.\end{aligned}$$
Hence, sum of first 20 terms = 2520.