The standard results for the sum of r, r squared and r cubed, using them to sum polynomial expressions in r, splitting sums by linearity, and adjusting limits.

What this dot point is asking

OCR wants you to know and use the standard summation results for $\sum r$, $\sum r^2$ and $\sum r^3$, to sum a polynomial expression in $r$ by splitting it with the linearity of summation, to factorise and simplify the resulting expression, and to adjust the limits when a sum starts above $1$ by writing it as a difference of two sums from $1$.

The standard results

These three formulae, together with $\sum 1 = n$, are the toolkit for the whole topic. They are quoted in the OCR formulae booklet, but you should know them well enough to apply them instantly.

Notice the neat fact that $\sum r^3 = \left(\sum r\right)^2$, which OCR sometimes asks you to verify or use.

Splitting a sum by linearity

Summation distributes over addition and pulls out constants, so a sum of a polynomial in $r$ breaks into separate standard sums. This is the routine first move on every summation question.

Factorising the result

After applying the formulae you almost always have a common factor of $n(n+1)$ (and a fractional coefficient). Taking out the largest common factor and tidying the bracket gives the clean factorised answer that the mark scheme expects. The reliable routine is: first write each standard sum in full, then identify the common factor shared by every term (it is usually $\tfrac{1}{6}n(n+1)$ when a $\sum r^2$ is present, or $\tfrac{1}{4}n^2(n+1)^2$ when the sum is dominated by $\sum r^3$), pull it out front, and simplify the bracket that remains into a single polynomial. Expanding the bracket, collecting like terms, and then refactoring often reveals a tidy linear or quadratic factor such as $(2n + 5)$ or $(n + 5)$. Always finish by checking your factorised formula against a small value of $n$ (for example $n = 1$ or $n = 2$) computed directly from the original sum; if the two agree, the algebra is almost certainly correct, and this quick check catches most slips.

Adjusting the limits

When a sum does not start at $r = 1$, rewrite it as the difference of two sums that do. This lets you use the standard formulae, which are stated from $1$.

The standard sums underpin the method of differences and are themselves provable by induction, tying the whole series and proof strand together.

Try this

Q1. Find $\displaystyle\sum_{r=1}^{n} (r + 2)$. [2 marks]

• Cue. $\sum r + \sum 2 = \tfrac{1}{2}n(n+1) + 2n = \tfrac{1}{2}n(n+5)$.

Q2. State the value of $\displaystyle\sum_{r=1}^{10} r^3$. [2 marks]

• Cue. $\tfrac{1}{4}(10)^2(11)^2 = \tfrac{1}{4}\cdot 100\cdot 121 = 3025$.