Arithmetic series

The sum of an arithmetic sequence is called the arithmetic series (a series is the sum of a sequence).

For an arithmetic sequence u_n=a+(n-1)d:

• S_n=\sum^n_{r=1}a+(r-1)d
  • S_n is the sum of the first n terms.
• The sum of the sequence is \frac n2(a+l), where:
  • n is the number of terms in the sequence.
  • a is the first term.
  • l is the last term.
• We can substitute the last term for the equation of the series, so:
  • \Sigma^n_{r=1}\frac n2(a+l)
  • u_n=a+(n-1)d
  • l=a+(n-1)d
  • \Sigma^n_{r=1}\frac n2(a+a+(n-1)\,d)
  • \Sigma^n_{r=1}\frac n2(2a+(n-1)\,d)

Useful formulae

As we derived above:

\Sigma^n_{r=1}\frac n2(2a+(n-1)\,d)