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How many terms are there in the A.P.?
$-1, -\frac{5}{6}, -\frac{2}{3}, -\frac{1}{2}, …….., \frac{10}{3}$.
Given:
Given A.P. is $-1, -\frac{5}{6}, -\frac{2}{3}, -\frac{1}{2}, …….., \frac{10}{3}$.
To do:
We have to find the number of terms in the given A.P.
Solution:
Here,
$a_1=-1, a_2=-\frac{5}{6}, a_3=-\frac{2}{3}$
Common difference $d=a_2-a_1=-\frac{5}{6}-(-1)=-\frac{5}{6}+1=\frac{-5+1\times6}{6}=\frac{1}{6}$
Let $\frac{10}{3}$ be the nth term.
We know that,
nth term $a_n=a+(n-1)d$
Therefore,
$a_{n}=-1+(n-1)(\frac{1}{6})$
$\frac{10}{3}=-1+\frac{n-1}{6}$
$\frac{10}{3}+1=\frac{n-1}{6}$
$\frac{10+1\times3}{3}=\frac{n-1}{6}$
$2(13)=n-1$ (On cross multiplication)
$n=26+1$
$n=27$
Hence, there are 27 terms in the given A.P.     
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