GCE Mathematics, A2 Unit 3 Pure Mathematics BTopic 3.4 - Sequences and series

The binomial theorem - recap from AS Unit 1

The binomial expansion of (a + b)n, for n a positive integer, is given by

(a + b)n = an + n1( )an–1b + (n

2)an–2b2 + … + ( nn -1 )abn–1 + bn

This is given in the Formula Booklet. Note that as the powers of a decrease, the powers of b increase, and there will be n + 1 terms in the expansion.

nr( ) = nCr = n!

r!(n – r)!

n! means n factorial remember 0! = 1

Examples

5! = 5 × 4 × 3 × 2 × 1 = 120 6C4 = 6!

4!2! = 1573( )= 7!

3!4! = 35

Pascal’s triangle can also be used to determine the coefficients in a binomial expansion.

Binomial expansion for negative and fractional values of n

The binomial expansion of (a + bx)n, where n is negative or a fraction, is given by

(a + bx)n = an + n1( )an–1bx + (n

2)an–2(bx)2 + …

which is only valid for |bxa | < 1.

Replacing a and b by 1, gives

(1 + x)n = 1 + nx + n(n – 1)2! x2 + n(n – 1)(n – 2)

3! x3 + …

valid for |x| < 1. This is given in the Formula Booklet.

(1 + x)–6 = 1 – 6x – 6(-7)2!  x2 – 6(-7)(-8)

3!  x3 + …

= 1 – 6x + 21x2 – 56x3 + … valid for |x| < 1

(1 – 4x)–3 = 1 – 3(–4x) – 3(-4)2! (–4x)2 – 3(-7)(-5)

3! (–4x)3 + …

= 1 + 12x + 96x2 + 640x3 + … valid for |4x| < 1

Arithmetic sequences

In an arithmetic sequence, consecutive terms differ by a fixed amount known as the common difference.

The terms can be written as t1 = a, t2 = a + d, t3 = a + 2d, etc, where a is the first term, and d is the common difference.

The formula for the nth term is given by tn = a + (n – 1)d.

The proof of the sum of an arithmetic sequence (Sn) could be asked for in an examination.

Sn = a + (a + d ) + (a + 2d ) + … + (a + (n – 1)d ) Sn = (a + (n – 1)d ) + (a + (n – 2)d ) + … + (a + d) + d

2Sn = (2a + (n – 1)d ) + (2a + (n – 1)d ) + … + (2a + (n – 1)d ) 2Sn = n(2a + (n – 1)d )

Sn = n2(2a + (n – 1)d )

The summation sign ⅀ can also be used.

Geometric sequences

In a geometric sequence, the ratio of consecutive terms is a fixed number known as the common ratio.

The terms can be written as t1 = a, t2 = ar, t3 = ar2 etc, where a is the first term, and r is the common ratio.

The formula for the nth term is given by tn = ar n–1.

The proof of the sum of a geometric sequence (Sn) could be asked for in an examination.

Sn = a + ar + ar2 + ar3 + … + arn–1

rSn = ar + ar2 + ar3 + ar4 +… + arn

rSn - Sn = arn – a

Sn = a (rn – 1)(r – 1) or Sn = a(1 – rn)

(1 – r) , r ≠ 1

When |r| < 1, rn → 0 as n → ∞, giving S∞ = a1 – r.

Periodic sequences

A periodic sequence is generated by repeating a set of numbers. For example, 1, 2, 3, 4, 1, 2, 3, 4, 1, …

Recurrence sequences xn+1 = f(xn)

A recurrence sequence is one in which successive terms are generated from a function involving the preceding terms.

Increasing/decreasing sequences

A sequence is increasing if each term is greater than the preceding term, and it is decreasing if each term is less than the preceding term.