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Sequences and series formulae

The 9 results the sequences and series notes display, each with the conditions it needs to be true and a link to the lesson that states it. Of the 9, the formulae booklet prints 6, 2 are yours to memorise and 1 is a working line the notes derive in front of you.

Years 12-13 · 4 lessons with formulae.

A result the Edexcel formulae booklet prints is marked as such; one it does not print is marked to learn; and a result badged derived in the lesson is a working line the notes build in front of you rather than a standard result to memorise.

4 lessons

The binomial expansion

nCr=n!r!(n−r)!^{n}C_{r} = \frac{n!}{r!(n − r)!}IN THE FORMULAE BOOKLETConditionsn and r non-negative integers with r ≤ n.
(a+b)n=an+nC1an−1b+…+bn(a + b)^{n} = a^{n} + \,^{n}C_{1}a^{n−1}b + … + b^{n}IN THE FORMULAE BOOKLETConditionsn a positive integer, so that the expansion terminates.

The general binomial expansion

(1+x)n=1+nx+n(n−1)2!x2+…(1 + x)^{n} = 1 + nx + \frac{n(n − 1)}{2!}x^{2} + …IN THE FORMULAE BOOKLETConditions|x| < 1 unless n is a non-negative integer, in which case the expansion terminates and holds for every x.
(a+bx)n=an(1+bx/a)n(a + bx)^{n} = a^{n}(1 + bx/a)^{n}NOT IN THE BOOKLET: DERIVED IN THE LESSONConditionsa ≠ 0, and the expansion that follows is valid only for |bx/a| < 1.

Arithmetic series

un=a+(n−1)du_{n} = a + (n − 1)dNOT IN THE BOOKLET: LEARN ITConditionsAn arithmetic sequence with common difference d; n a positive integer.
Sn=n2(2a+(n−1)d)=n2(a+l)S_{n} = \frac{n}{2}(2a + (n − 1)d) = \frac{n}{2}(a + l)IN THE FORMULAE BOOKLETConditionsAn arithmetic series, with l the nth term.

Geometric series

un=arn−1u_{n} = ar^{n−1}NOT IN THE BOOKLET: LEARN ITConditionsA geometric sequence with common ratio r; n a positive integer.
Sn=a(1−rn)1−rS_{n} = \frac{a(1 − r^{n})}{1 − r}IN THE FORMULAE BOOKLETConditionsr ≠ 1. When r = 1 the sum is simply na.
S∞=a1−rS_{∞} = \frac{a}{1 − r}IN THE FORMULAE BOOKLETConditions|r| < 1. Outside that the series diverges and there is no sum to infinity.

Every formula in the library is indexed by unit on the equation index, and the sequences and series notes are where these results are derived and used. The 2 the booklet does not print are on the list of every formula to memorise, alongside the rest of the course.

Also: Coordinate geometry formulae · Trigonometry formulae