Summation of arithmetic sequence

Question Sample Titled 'Summation of arithmetic sequence'

Let ${a},{d}$ and ${l}$ be the first term, common difference and the ${n}$th term of an arithmetic sequence respectively, then the sum of the first ${n}$ terms of the arithmetic sequence is given by:

${S}{\left({n}\right)}=\dfrac{{n}}{{2}}{\left({a}+{l}\right)}$

or can be written as:

${S}{\left({n}\right)}=\dfrac{{n}}{{2}}{\left[{2}{a}+{\left({n}-{1}\right)}{d}\right]}$

The sum of an arithmetic sequence is also called an arithmetic series.

Furthermore, we can extract the ${n}$-th term of the arithmetic sequence from the arithmetic series:

${T}{\left({n}\right)}={S}{\left({n}\right)}-{S}{\left({n}-{1}\right)}$

Note that the first term of the arithmetic sequence equals to that of the arithmetic series:

${T}{\left({1}\right)}={S}{\left({1}\right)}$

 Find the sum of the arithmetic series ${10}+{8}+{6}+$ ... to ${20}$ terms.

 Let ${a}$ , ${d}$ and ${n}$ be the first term, the common difference and the number of terms taken respectively. ∵ ${a}={10}$ , ${n}={20}$ and ${d}={8}-{10}=-{2}$ ∴ ${S}{\left({20}\right)}$  $=\dfrac{{20}}{{2}}{\left[{2}{\left({10}\right)}+{\left({20}-{1}\right)}{\left(-{2}\right)}\right]}$  $=-{180}$

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