4 Sequences and series — notes on one page

4.1 · Understand and use the binomial expansion of (a + bx)ⁿ for positive integer n; the notations n! and nCr; link to binomial probabilities; extend to any rational n, including for approximation, valid for |bx/a| < 1.

Definition. Specification 4.1: understand and use the binomial expansion of $(a+bx)^n$ for positive integer $n$; the notations $n!$ and $\binom{n}{r}$; link to binomial probabilities; extend to any rational $n$, including its use for approximation; be aware that the expansion is valid for $\left|\dfrac{bx}{a}\right|<1$.

• For a positive integer $n$, $(a+bx)^n=\sum\binom{n}{r}a^{n-r}(bx)^r$, and the same coefficients give binomial probabilities $\binom{n}{r}p^r(1-p)^{n-r}$. For a rational or negative $n$ use $(1+x)^n=1+nx+\dfrac{n(n-1)}{2!}x^2+\dfrac{n(n-1)(n-2)}{3!}x^3+\cdots$, an infinite series valid only for $|x|<1$.
• Replace $x$ by the whole of $bx$ in brackets, including its sign and coefficient, so $(1-9x)^{1/2}$ has second term $\tfrac12(-9x)$ and third $\dfrac{\frac12\cdot(-\frac12)}{2}(-9x)^2$.
• When the first term is not $1$, take it out first: $\sqrt{4+5x}=2(1+\tfrac54x)^{1/2}$, and the validity becomes $\left|\tfrac54x\right|<1$.
• State the range of validity, $|x|<\dfrac{1}{9}$ or $|9x|<1$, and check a substituted value against it before using the series for an approximation; a value outside the range gives no approximation at all.
• Simplify each coefficient fully.

What the examiner wants. Brackets round the whole term and a stated range. The June 2024 Paper 1 report says the binomial structure was generally very good but inconsistency with brackets and not simplifying cost accuracy marks, the most common error being $9$ instead of $-9$; on validity, many could not state $|x|<\frac19$ and misread the modulus, writing $x>\frac19$.

Exam tip. For a rational-power expansion, factor the constant first and state the validity condition from the transformed bracket.

4.2 · Work with sequences including those given by a formula for the nth term and those generated by a simple relation of the form xₙ₊₁ = f(xₙ); increasing sequences; decreasing sequences; periodic sequences.

• An explicit rule gives $x_n$ directly from $n$, whereas a recurrence relation defines each term from one or more preceding terms and needs an initial value.
• A sequence is increasing when $x_{n+1}>x_n$ and decreasing when $x_{n+1}<x_n$ throughout the stated range of $n$.
• A sequence is periodic if its terms repeat after a fixed positive number of steps; the least such number is its period.
• When using a recurrence, retain sufficient accuracy between steps and check any invariant interval.
• A common error is to apply $f$ repeatedly to the initial term instead of to the latest term.

Exam tip. Generate enough consecutive terms to demonstrate the claimed monotonicity or period rather than guessing from the first change.

4.3 · Understand and use sigma notation for sums of series.

• The notation $\sum_{r=a}^{b}f(r)$ means add the values of $f(r)$ for each integer $r$ from $a$ to $b$ inclusive.
• To write a series in sigma notation, identify a formula for its general term together with the correct first and last index values.
• Use linearity to split sums: for example, $\sum_{r=1}^{n}(ar+b)=a\sum_{r=1}^{n}r+b\sum_{r=1}^{n}1$, with $\sum_{r=1}^{n}1=n$.
• Check the endpoint terms after changing an index.
• A common error is to treat the upper limit as the number of terms when the lower limit is not $1$.

Exam tip. Check a sigma answer by substituting both endpoint indices and confirming the number of generated terms.

4.4 · Understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms.

• An arithmetic sequence has constant difference $d$ and nth term $u_n=a+(n-1)d$, where $a$ is the first term.
• The sum of its first $n$ terms is $S_n=n[2a+(n-1)d]/2$, equivalently $S_n=n(a+l)/2$ when the last term $l$ is known.
• Pairing the first and last terms, then the second and penultimate terms, gives $n$ equal pairs of total $a+l$ across two copies of the series, proving $S_n=n(a+l)/2$.
• The term number $n$ must be a positive integer.
• A common error is to use $a+nd$ for the nth term, shifting every term by one difference.

Exam tip. Find the integer term count from the nth-term formula before applying the arithmetic sum formula.

4.5 · Understand and work with geometric sequences and series, including the formulae for the nth term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |r| < 1; modulus notation.

• A geometric sequence has constant ratio $r$ and nth term $u_n=ar^{n-1}$, where $a$ is the first term.
• For $r\ne1$, subtracting $rS_n$ from $S_n$ proves $S_n=a(1-r^n)/(1-r)$; an equivalent form may be more convenient when $r>1$.
• A geometric series has a finite sum to infinity only when $|r|<1$, in which case $S_\infty=a/(1-r)$; for a finite-sum threshold, isolate $r^n$ and use logarithms.
• A negative ratio makes term signs alternate, but convergence still depends on its modulus.
• A common error is to use $r<1$ instead of $|r|<1$, which would wrongly accept ratios below $-1$.

Exam tip. State the common ratio and verify convergence before calculating an infinite geometric sum.

4.6 · Use sequences and series in modelling.

• A constant additive change suggests an arithmetic model, while a constant multiplier or percentage change suggests a geometric model.
• State what the term number represents and whether the initial value is $u_1$ or $u_0$ before forming a term or sum.
• Repeated deposits, withdrawals or other fixed adjustments can be represented by a recurrence and often rewritten using a finite geometric sum.
• Interpret results within the context and identify assumptions such as fixed rates or indefinite continuation.
• A common error is to use an infinite sum when the process has not converged or has a finite stopping point.

Exam tip. Build the distance model by separating the first drop from the two-way geometric rebound distances.