How do you derive \sum{r=1}^{n}r?

Write the sum forwards and backwards: each pair adds to n+1, there are n pairs, so 2S=n(n+1).

Is \sum r^2=\left(\sum r\right)^2?

No. The sum of squares is \frac{n(n+1)(2n+1)}{6}, not the square of \frac{n(n+1)}{2}.

How do you sum \sum{r=1}^{n}r(r+2)?

Expand to \sum r^2+2\sum r, substitute the standard results, then factorise.

How do you find \sum{r=a}^{b}f(r)?

\sum{r=1}^{b}f(r)-\sum{r=1}^{a-1}f(r)

Is the method of differences needed in this topic?

No: only the standard results and expanding are required.

How do you check a summation formula?

Substitute n=1 and n=2 and compare with adding the first terms directly.

Summation of finite seriesEdexcel International A Level Further Maths: Flashcards

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Value of $\sum_{r=1}^{n}c$, where $c$ is a constant?

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Value of $\sum_{r=1}^{n}c$ , where $c$ is a constant?: $cn$
State $\sum_{r=1}^{n}r$ .: $\frac{n(n+1)}{2}$
State $\sum_{r=1}^{n}r^2$ .: $\frac{n(n+1)(2n+1)}{6}$
How do you derive $\sum_{r=1}^{n}r$ ?: Write the sum forwards and backwards: each pair adds to $n+1$ , there are $n$ pairs, so $2S=n(n+1)$ .
Is $\sum r^2=\left(\sum r\right)^2$ ?: No. The sum of squares is $\frac{n(n+1)(2n+1)}{6}$ , not the square of $\frac{n(n+1)}{2}$ .
How do you sum $\sum_{r=1}^{n}r(r+2)$ ?: Expand to $\sum r^2+2\sum r$ , substitute the standard results, then factorise.
Find $\sum_{r=1}^{n}r(r+2)$ .: $\frac{n(n+1)(2n+7)}{6}$
Find $\sum_{r=1}^{n}(2r+3)$ .: $n(n+4)$
How do you find $\sum_{r=a}^{b}f(r)$ ?: $\sum_{r=1}^{b}f(r)-\sum_{r=1}^{a-1}f(r)$
Which common factor appears when combining $\sum r^2$ and $\sum r$ ?: $\frac{n(n+1)}{6}$
Is the method of differences needed in this topic?: No: only the standard results and expanding are required.
How do you check a summation formula?: Substitute $n=1$ and $n=2$ and compare with adding the first terms directly.

Exam questions on Summation of finite series

A sequence has $r$ th term $u_r=2r+3$ , and $S_n$ is the sum of the first $n$ terms.
Find the smallest value of $n$ for which $S_n>1000$ .2 marks
Let $S_n=1\times3+2\times4+3\times5+\dots+n(n+2)$ , the sum of the first $n$ terms of a series.
Find the sum of the 6th to the 10th terms of the series, inclusive.2 marks
An orange display is built in layers. Layer $r$ , counting from the top with $r=1,2,3,\dots$ , is a square arrangement containing $r^2$ oranges.
Find the number of oranges in layers $13$ to $20$ inclusive.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).