Property 1
\(n\in\mathbb{N}^{+}\)
- \(1+2+3+4+\ldots+n=\samefrac{n\cdot(n+1)}{2}\)
- \(1+3+5+7+\ldots+(2n-1)=n^2\)
- \(2+4+6+8+\ldots+(2n)=n(n+1)\)
The sum of consecutive numbers with a constant order of increase is determined with the help of the following two formulas:
\(\text{Number of Terms}=\samefrac{\text{Last Term}-\text{First Term}}{\text{Amount of Increase}}+1\)
\(\text{Sum}=\samefrac{(\text{Last Term}+\text{First Term})\cdot\text{Number of Terms}}{2}\)
Example
\(\text{Number of Terms}=\samefrac{46-7}{3}+1=14\)
\(\text{Total}=\samefrac{(46+7)\cdot14}{2}=371\)
1.
\(1+2+3+4+\ldots+19=?\)
2.
\(1+2+3+\ldots+30=?\)
3.
\(10+11+12+\ldots+18=?\)
4.
\(13+14+15+\ldots+21=?\)
5.
\(1+2+3+\ldots+x=55\)
\(\Rightarrow x=?\)
6.
\(1+2+3+\ldots+x=45\)
\(\Rightarrow x=?\)
7.
\(7+8+9+\ldots+x=70\)
\(\Rightarrow x=?\)
8.
\(2+4+6+\ldots+20=?\)
9.
\(10+12+14+\ldots+28=?\)
10.
\(12+14+16+\ldots+24=?\)
11.
\(2+4+6+8+\ldots+x=420\)
\(\Rightarrow x=?\)
12.
\(2+4+6+8+\ldots+x=156\)
\(\Rightarrow x=?\)