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Q.
Find the sum of all odd integers between $2$ and $100$ divisible by $3$.
Sequences and Series
Solution:
The odd integers between $2$ and $100$ which are divisible by $3$ are $3, 9, 15, 21,.... 99$. Clearly, it is an $A.P$. with first term $a = 3$ and common difference $d = 6$. Let there be n terms in this sequence. Then,
$ a_{n} = 99$
$ \Rightarrow 3 + (n - 1) \times 6 = 99$
$ \Rightarrow n = 17$
$ \therefore $ Required sum $= \frac{n}{2} \left[a +l\right]$
$ =\frac{17}{2} \left[3 + 99\right]$
$= 867$.