Find the number of terms in each of the following APs:
(i) $7, 13, 19, …, 205$
(ii) $18, 15\frac{1}{2}, 13, …, -47$

To do:

We have to find the number of terms in each of the given APs.

Solution:

(i) Given A.P. is $7, 13, 19, …, 205$.

Here,

$a_1=7, a_2=13, a_3=19$

Common difference $d=a_2-a_1=13-7=6$

Let $205$ be the nth term.

We know that,

nth term $a_n=a+(n-1)d$

Therefore,

$a_{n}=7+(n-1)(6)$

$205=7+n(6)-1(6)$

$205-7=6n-6$

$198+6=6n$

$6n=204$

$n=\frac{204}{6}$

$n=34$

Hence, there are 34 terms in the given A.P.     

(ii) Given A.P. is $18, 15\frac{1}{2}, 13, …, -47$.

Here,

$a_1=18, a_2=15\frac{1}{2}, a_3=13$

Common difference $d=a_2-a_1=15\frac{1}{2}-18=\frac{15\times2+1}{2}-18=\frac{31-2\times18}{2}=\frac{31-36}{2}=\frac{-5}{2}$

Let $-47$ be the nth term.

We know that,

nth term $a_n=a+(n-1)d$

Therefore,

$a_{n}=18+(n-1)(\frac{-5}{2})$

$-47=18+\frac{-5(n-1)}{2}$

$-47-18=\frac{-5n+5}{2}$

$-65=\frac{-5n+5}{2}$

$2(-65)=-5n+5$       (On cross multiplication)

$5n=130+5$

$5n=135$

$n=\frac{135}{5}$

$n=27$

Hence, there are 27 terms in the given A.P.       

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Updated on: 10-Oct-2022

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