Arithmetic Progressions

Stage: 3 and 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This problem follows on from the article Clever Carl. In the article we described Gauss's method of calculating the progression $1+2+\ldots +n$ and came up with the formula $$1+2+\ldots n = \frac{1}{2}n(n+1)$$

(a) Calculate $2+4+\ldots +2n$.
Show that $1+3+5+\ldots +(2n-1)=n^2$.


The sum $1+2+3+\ldots +n$ in an example of an arithmetic progression. In general, an arithmetic progression is a sum of the form $a+(a+d)+(a+2d)+\ldots +(a+nd)$ where $a$, $d$ and $n$ are integers.

(b) Prove that $a+(a+d)+(a+2d)+\ldots +(a+nd)=\frac{1}{2}(n+1)(2a+nd)$.

(c) Find the sum of all the integers less than $1000$ which are not divisible by $2$ or $3$.


search engine page

Published February 2010.