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A Levels / Maths

Sequences and Series

Master A-Level sequences and series, including binomial expansion, nth term formulas, arithmetic sequences, geometric sequences, sigma notation, finite and infinite series, and sum to infinity.

The binomial expansion expands expressions of the form (a+bx)n(a+bx)^n, and for a positive integer nn the binomial expansion formula uses binomial coefficients to generate each term.
The binomial theorem formula can be written as (a+b)n=∑r=0n(nr)an−rbr(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r, where (nr)=n!r!(n−r)!\binom{n}{r}=\frac{n!}{r!(n-r)!}.
A binomial expansion equation uses factorial notation and nCrnCr coefficients to determine the coefficients and powers of each term in an expansion.
The binomial theorem can also be extended to rational values of nn, allowing binomial expansion to be used for approximation when the required convergence condition is satisfied.
For expansions such as (a+bx)n(a+bx)^n, the extended binomial expansion is valid when ∣bxa∣<1\left|\frac{bx}{a}\right|<1, which is important when using the expansion for approximations.
The nth term describes the value of a sequence at position nn, and an nth term formula allows any required term to be calculated without listing all previous terms.
To understand how to find the nth term, identify how the terms change and express the pattern as a formula of the nth term in terms of nn.
Sequences may be defined directly by an nth term formula or recursively using a relation such as xn+1=f(xn)x_{n+1}=f(x_n), and they may be increasing, decreasing or periodic.
Arithmetic sequences have a constant common difference dd, with nth term an=a+(n−1)da_n=a+(n-1)d, making the arithmetic sequence formula useful for finding any term.
Arithmetic series and sequences are connected because an arithmetic series is the sum of the terms of an arithmetic sequence, with arithmetic series formula Sn=n2[2a+(n−1)d]S_n=\frac{n}{2}[2a+(n-1)d].
The sum of arithmetic sequence terms can also be written as Sn=n2(a+l)S_n=\frac{n}{2}(a+l), where aa is the first term and ll is the final term.
Geometric sequences have a constant common ratio rr, and the geometric sequence formula for the nth term is an=arn−1a_n=ar^{n-1}.
For geometric sequence summation, the sum of the first nn terms is Sn=a(1−rn)1−rS_n=\frac{a(1-r^n)}{1-r} for r≠1r\ne1; this is the standard formula for the summation of geometric sequence terms.
An infinite geometric sequence converges when ∣r∣<1|r|<1, and the sum to infinity formula is S∞=a1−rS_\infty=\frac{a}{1-r}, which gives the sum of geometric series that converge.
Sigma notation uses the summation symbol ∑\sum, also called the symbol for summation, to represent sums compactly and allows sequences and series to be used in mathematical modelling and real-world problems.
Sequences and Series | Binomial Expansion & Nth Term | TSL