Question
Find the sum of $7 + 10\frac{1}{2} + 14..... + 84$.
✓
Answer
Clearly, the terms of the given sum form an A.P,
with, a = 7
$d = 10\frac{1}{2} - 7 = 3\frac{1}{2} = \frac{7}{2}$
Let the number of terms of the AP be n.
We know that
l = a + (n - 1)d
$ \Rightarrow 84 = 7 + (n - 1)\frac{7}{2}$
$ \Rightarrow (n - 1)\frac{7}{2} = 84 - 7$
$ \Rightarrow (n - 1)\frac{7}{2} = 77$
$ \Rightarrow (n - 1) = 22$
$ \Rightarrow n = 22 + 1$
$ \Rightarrow n = 23$
Again, we know that
${S_n} = \frac{n}{2}(a + l)$
$ \Rightarrow {S_{23}} = \frac{{23}}{2}(7 + 84)$
$ \Rightarrow {S_{23}} = \frac{{2093}}{2}$
$ \Rightarrow {S_{23}} = \frac{{2093}}{2}$
$ \Rightarrow {S_{23}} = 1046\frac{1}{2}$
Hence, the required sum is $1046\frac{1}{2}$
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