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Question
How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\]
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Solution
\[- 1, - \frac{5}{6}, - \frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}\]
Here, we have:
a =\[- 1\]
\[d = \left( \frac{- 5}{6} - \left( - 1 \right) \right) = \left( 1 - \frac{5}{6} \right) = \frac{1}{6}\]
\[ a_n = \frac{10}{3}\]
Let there be n terms in the given A.P.
\[\text { Also }, a_n = a + \left( n - 1 \right)d\]
\[ \Rightarrow \frac{10}{3} = - 1 + \left( n - 1 \right)\frac{1}{6}\]
\[ \Rightarrow \frac{13}{3} = \left( n - 1 \right)\frac{1}{6}\]
\[ \Rightarrow 26 = \left( n - 1 \right)\]
\[ \Rightarrow 27 = n\]
Thus, there are 27 terms in the given A.P.
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