Question

Find the sum of a given arithmetic sequence:
\(\displaystyle\sum\ {n}={1}\ {100}\ {\left({6}\ -\ {12}\ {n}\right)}\)

Answer 1

Formula used:
a) \(\displaystyle\sum\ {k}={1}\ {n}{k}={n}\ {2}\ {\left({n}\ +\ {1}\right)}\)
b) \(\displaystyle\sum\ {k}={1}\ {n}\ {c}={c}{n}\)
Calculation:
In order to find the sum of an arithmetic sequence, you need
\(\displaystyle{S}\ {n}=\sum\ {n}={1}\ {100}\ {\left({6}\ -\ {12}{n}\right)}\)
Write \(\displaystyle\sum\ {n}={1}\ {100}\ {\left({6}\ -\ {12}{n}\right)}\) as follows,
\(\displaystyle\sum\ {n}={1}\ {100}\ {\left({6}\ -\ {12}{n}\right)}=\sum\ {n}={1}\ {100}\ {6}\ -\ {12}\ \sum\ {n}={1}\ {100}\ {n}\)
Now to find \(\displaystyle\sum\ {n}={1}\ {100}\ {6},\)
Substitute the values of n in equation (b).
We get,
\(\displaystyle\sum\ {n}={1}\ {100}\ {6}\)
\(\displaystyle={6}\ {\left({100}\right)}\)
\(\displaystyle={600}\)
Now to find the \(\displaystyle{12}\ \sum\ {n}={1}\ {100}\ {n}\)
Substitute the values of n in equation (a).
We get,
\(\displaystyle{12}\ \sum\ {n}={1}\ {100}\ {n}\)
\(\displaystyle={12}\ {\left({100}{\left({100}\ +\ {1}\right)}{2}\right)}\)
\(\displaystyle={\left({100}\ \times\ {101}\ {4}\right)}\)
\(\displaystyle={2525}\)
Then,
\(\displaystyle\sum\ {n}={1}\ {100}\ {6}\ -\ {12}\ \sum\ {n}={1}\ {100}\ {n}={600}\ -\ {2525}\)
\(\displaystyle=\ -{1925}\)
Hence
\(\displaystyle\sum\ {n}={1}\ {100}{\left({6}\ -\ {12}{n}\right)}\ =\ -{1925}\)