Question

Find the number of terms in each of the following A.P.

1. $7, 13, 19, ……, 205$
2. $18,15 \frac 12 ,13,……,−47$

Answer 1

(i) $7, 13, 19, ..…, 205$
For this A.P., $a = 7$ and $d = a_2 − a_1 = 13 − 7 = 6$
Let there are $n$ terms in this A.P. $a_n = 205$
We know that $a_n = a + (n − 1) d$
Therefore, $205 = 7 + (n − 1) 6$
$198 = (n − 1) 6$
$33 = (n − 1)$
$n = 34$

Therefore, this given series has 34 terms in it.

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(ii) $18,15\frac 12 ,13, ..…,−47$ For this A.P.,
$a = 18$
$d = a_2-a_1$
$d= 15 \frac 12 -18$

$d = \frac {31}{2} - 18$

$d= \frac {31-36}{2}= -\frac 52$
Let there are $n$ terms in this A.P.
Therefore, $a_n = −47$and we know that,
$a_n = a +(n-1)d$
$-47 = 18 + (n-1)(-\frac 52)$

$-47-18 = (n-1)(-\frac 52)$

$-65 = (n-1)(-\frac 52)$

$\frac {-65 \times 2}{-5} = n-1$

$n-1 = \frac {-130}{-5}$
$n-1 = 26$
$n = 27$

Therefore, this given A.P. has 27 terms in it.