Pure: Sequences and Series (incl. Binomial Expansion)
Sequences and series covers arithmetic and geometric progressions, including their nth term and sum formulae and the sum to infinity of a convergent geometric series, alongside the binomial expansion of expressions such as (1+x)^n for positive integer and general powers of n. It is used to model growth and decay, such as savings, population and depreciation.
Method
- Decide whether a sequence is arithmetic (constant common difference d) or geometric (constant common ratio r) by checking consecutive term differences or ratios.
- Use u_n = a + (n-1)d for arithmetic sequences and u_n = a*r^(n-1) for geometric sequences to find any term.
- Use S_n = (n/2)(2a + (n-1)d) for arithmetic sums, or S_n = a(1-r^n)/(1-r) for geometric sums; use S_infinity = a/(1-r) only when |r| < 1.
- For (1+x)^n with n a positive integer, use Pascal's triangle or nCr coefficients; for fractional or negative n, use the general binomial series and state the range of x for which it is valid.
- In (a+bx)^n expansions, take out a factor of a^n first if needed so the bracket starts with 1, then expand using the general term.
- To find a specific term (such as the coefficient of x^3 or the term independent of x), write the general term of the expansion and set the power of x equal to the required value.
Worked example
A geometric sequence has first term a = 5 and common ratio r = 1.2. Find the 10th term of the sequence, and the sum of the first 10 terms, giving both answers to 3 significant figures.
- Use u_n = a*r^(n-1) with a=5, r=1.2, n=10: u_10 = 5 x 1.2^9.
- Calculate 1.2^9 = 5.15978 (5dp), so u_10 = 5 x 5.15978 = 25.7989, which rounds to 25.8 (3sf).
- Use S_n = a(1-r^n)/(1-r) with n=10: S_10 = 5(1 - 1.2^10)/(1 - 1.2).
- Calculate 1.2^10 = 6.19174 (5dp), so S_10 = 5(1 - 6.19174)/(-0.2) = 5 x 25.9587.
- Evaluate: S_10 = 129.793, which rounds to 130 (3sf).
- Final answer: u_10 = 25.8 (3sf), S_10 = 130 (3sf).
Practice questions
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Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
An arithmetic sequence has first term 6 and common difference d. Given that the 20th term is 63, find the value of d and the sum of the first 20 terms.
The first three terms of a geometric sequence, in order, are k, k+4 and 4k, where k is a positive constant. Show that 3k^2 - 8k - 16 = 0, and hence find the value of k.
f(x) = (1 + 4x)^(1/2). Find the binomial expansion of f(x) in ascending powers of x up to and including the term in x^2, simplifying each coefficient and stating the range of values of x for which the expansion is valid. Hence estimate sqrt(1.08) to 4 decimal places.
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