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.