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Sums of Natural Numbers

What are Series

Series
  • A series is a sum of a finite or infinite number of terms
  • \(\sum\) is the notation used to define a series
  • E.G. \(\(\sum_{r=1}^{5} (2r+1)\)\)
  • The answer to this would be \(3+5+7+9+11 = 35\)

Essential Formulae

The formulae
  • n: \(\(\sum_{r=1}^{n}1=n\)\)
  • Sum of the first n integers: \(\(\sum_{r=1}^{n}r=1+2+3+\dots+n = \frac{n\times (n+1)}{2}\)\)
  • Sum of the first n squares: \(\(\sum_{r=1}^{n}r^{2}=\frac{1}{6}n(n+1)(2n+1)\)\)
  • Sum of the first n cubes: \(\(\sum_{r=1}^{n}r^{3}=\frac{1}{4}n^{2}(n+1)^2\)\)
  • Subtracting series: \(\(\sum_{r=a}^nf(r)=\sum_{r=1}^nf(r)-\sum_{r=1}^{a-1}f(r)\)\)
  • Variables: \(\(\sum_{r=1}^nkf(r)=k\sum_{r=1}^nf(r)\)\)

Subtracting Series

Subtracting series
  • To add up all the integers from 3 to 9: \(\(\sum_{n=1}^9r-\sum_{r=1}^2r\)\)
  • This gives us \(\(\sum_{n=3}^9r\)\)
With algebra
  • Example with algebra: \(\(\sum_{r=n}^{3n}r=\sum_{r=1}^{3n}r-\sum_{r=1}^{n-1}r\)\)
  • \[=\frac{1}{2}(3n)(3n+1)-\frac{1}{2}(n-1)(n)\]
  • \[=\frac{1}{2}n(3(3n+1)-(n-1))\]
  • \[=\frac{1}{2}n(9n+3-n+1)\]
  • \[\frac{1}{2}n(8n+4)\]
  • \[2n(2n+1)\]

Breaking Up Summations

Breaking up summations
  • \[\sum_{i=1}^n(a_{i}+b{i})=a_{1}+b_{1}+a_{2}+b_{2}+\dots+a_{n}+b_{n}\]
  • \[=a_{1}+a_{2}+a_{3}+a_{4}+\dots+a_{n}+b_{1}+b_{2}+b_{3}+b_{4}+\dots+b_{n}\]
  • \[=\sum_{i=1}^na_{i}+\sum_{i=1}^nb_{i}\]
Show that \(\sum_{r=1}^n(3r+2)=\frac{n}{2}(3n+7)\)
  • \[\sum_{r=1}^n3r+\sum_{r=1}^n2\]
  • \[\frac{3}{2}n(n+1)+2n\]
  • \[\frac{n}{2}[3(n+1)+4]\]
  • \[\frac{n}{2}(3n+3+4)\]
  • \[\frac{n}{2}(3n+7)\]
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