Summing series of powers of integers, relationships between roots and coefficients of polynomials, transforming equations with new roots, and the method of differences.

What this dot point is asking

Edexcel wants you to use the standard results for $\sum r$, $\sum r^2$ and $\sum r^3$ to sum polynomial series, link the roots of a polynomial to its coefficients through symmetric functions, form new equations whose roots are transformed versions of the originals, and telescope sums using the method of differences. These techniques are bread-and-butter Core Pure and frequently set as "show that" questions, so every line of working must be visible.

Standard summation formulae

You build sums of any polynomial in $r$ from the three standard results, splitting the sum term by term and then factorising to a tidy closed form. Linearity lets you pull out constants and split across additions: $\sum (ar^2 + br + c) = a\sum r^2 + b\sum r + cn$.

The neat fact that $\sum r^3 = (\sum r)^2$ is worth remembering; it occasionally lets you check a long calculation.

Roots and coefficients

For a polynomial with roots $\alpha, \beta, \gamma, \dots$, the elementary symmetric functions (the sum of roots, the sum of products in pairs, and so on) are fixed by the coefficients. This lets you evaluate symmetric expressions in the roots without ever solving the polynomial.

Useful derived identities include $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ and, for cubics, $\alpha^2 + \beta^2 + \gamma^2 = (\sum\alpha)^2 - 2\sum\alpha\beta$.

Transforming roots

To find the equation whose roots are a transformation of the originals, substitute the inverse transformation. If the new roots are $y = 2\alpha$, then $\alpha = \frac{y}{2}$, so replace $x$ by $\frac{y}{2}$ in the original polynomial and clear fractions. The resulting polynomial in $y$ has exactly the transformed roots.

Method of differences

If each term of a sum can be written as $f(r) - f(r+1)$ (or $f(r-1) - f(r)$), the sum telescopes: interior terms cancel in adjacent pairs, leaving only contributions from the start and end. Standard exam terms such as $\frac{1}{r(r+1)}$ decompose by partial fractions into exactly this form. Always write out the first two and last two bracketed terms so you can see precisely which boundary terms survive.

Examples in context

This dot point underpins much of Core Pure. The standard summation results are proved by proof by induction, and the method of differences is itself a route to those proofs. Summing series of complex exponentials, treating $\sum\cos r\theta$ as the real part of a geometric series in $e^{i\theta}$, links to complex numbers and de Moivre. Roots and coefficients reappear whenever a question gives one complex root of a real polynomial and expects you to use the conjugate pair, and the transformed-roots technique is the standard way to build a new equation without solving the old one. Maclaurin and Taylor series (a later option) use similar coefficient-matching ideas.

Try this

Q1. Evaluate $\displaystyle\sum_{r=1}^{n} (2r - 1)$. [3 marks]

• Cue. $2\sum r - \sum 1 = n(n+1) - n = n^2$.

Q2. The quadratic $x^2 - 5x + 6 = 0$ has roots $\alpha, \beta$. Find $\alpha^2 + \beta^2$. [3 marks]

• Cue. $(\alpha + \beta)^2 - 2\alpha\beta = 25 - 12 = 13$.

Q3. Find a quadratic with roots $\alpha + 2$ and $\beta + 2$, where $\alpha, \beta$ are the roots of $x^2 - 6x + 8 = 0$. [4 marks]

• Cue. Substitute $x = y - 2$: $(y-2)^2 - 6(y-2) + 8 = y^2 - 10y + 24 = 0$.