Find the sum:$7 + 10\frac{1}{2} + 14 + ……… + 84$

Given:

Given sequence is $7 + 10\frac{1}{2} + 14 + ……… + 84$.

To do:

We have to find the sum of $7 + 10\frac{1}{2} + 14 + ……… + 84$.

Solution:

Here,

\( 7+10 \frac{1}{2}+14+\ldots+84 \) is in A.P.

\( a=7, d=10 \frac{1}{2}-7=3 \frac{1}{2}=\frac{7}{2} \) and \( l=84 \)

We know that,

\( a_{n}=a+(n-1) d \)

\( \Rightarrow 84=7+(n-1) \times \frac{7}{2} \)

\( \Rightarrow 84 \times 2=[7 \times 2+(n-1) 7] \)
\( 168=14+7 n-7 \)

\( \Rightarrow 7 n=168-14+7 \)
\( \Rightarrow 7 n=161 \)

\( \Rightarrow n=\frac{161}{7}=23 \)
\( \mathrm{S}_{n}=\frac{n}{2}[a+l] \)

\( =\frac{23}{2}[7+84] \)
\( =\frac{23}{2} \times 91 \)

\( =\frac{2093}{2} \)

Therefore, the sum of the given sequence is $\frac{2093}{2}$. 

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Updated on: 10-Oct-2022

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