4 Sequences and series — 9MA0 spec coverage

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.

For a positive integer $n$ , $(a+bx)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}(bx)^r$ , where $\binom{n}{r}=n!/[r!(n-r)!]$ , $n!=n(n-1)\cdots1$ and $0!=1$ ; Pascal's relation is $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}$ for $1\le r\le n-1$ .
The same coefficients appear in binomial probabilities: $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$ for $X\sim B(n,p)$ .
For rational $n$ , write the expression as $a^n(1+u)^n$ and use $1+nu+n(n-1)u^2/2!+\cdots$ , valid for $|u|<1$ .
For an approximation, choose a nearby convenient value and retain the requested number of terms. A common error is to omit powers of the coefficient $b$ from terms involving $(bx)^r$ .

Tier 1 · Easy

2 marks

ORIGINAL

Find the coefficient of $x^2$ in the expansion of $(1+4x)^5$ .

Tier 2 · Standard

4 marks

ORIGINAL

For $(1-2x)^{-1/2}$ , obtain the constant, $x$ and $x^2$ terms. Also give the interval of $x$ on which this series is valid.

Tier 3 · Hard

5 marks

ORIGINAL

Use the first three terms of a binomial expansion of $(64+x)^{1/3}$ to estimate $\sqrt[3]{65}$ . Give the estimate to $5$ decimal places and justify that the expansion is valid at the value of $x$ used.

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.

Tier 1 · Easy

2 marks

ORIGINAL

The sequence $(x_n)$ is defined by $x_n=7-3n$ for $n\ge1$ . Write down its first three terms and state whether it is increasing or decreasing.

Tier 2 · Standard

3 marks

ORIGINAL

A sequence is defined by $x_1=1$ and $x_{n+1}=5-x_n$ . Find $x_2,x_3,x_4,x_5$ and state its period.

Tier 3 · Hard

5 marks

ORIGINAL

A sequence is defined by $x_1=2$ and $x_{n+1}=(x_n+6)/2$ . Let $y_n=6-x_n$ . Show that $(y_n)$ is geometric, find a formula for $x_n$ , and hence show that $(x_n)$ is increasing.

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 $1$ .

Tier 1 · Easy

2 marks

ORIGINAL

Evaluate $\sum_{r=1}^{4}(2r+1)$ .

Tier 2 · Standard

2 marks

ORIGINAL

Write the series $5+9+13+\cdots+41$ using sigma notation.

Tier 3 · Hard

5 marks

ORIGINAL

Given that $\sum_{r=1}^{n}(4r-1)=210$ , find the positive integer $n$ . Show all stages of your working.

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.

Tier 1 · Easy

2 marks

ORIGINAL

Find the $18$ th term of the arithmetic sequence $11,17,23,\ldots$ .

Tier 2 · Standard

3 marks

ORIGINAL

An arithmetic series has first term $7$ , last term $151$ and common difference $8$ . Find the number of terms and the sum of the series.

Tier 3 · Hard

5 marks

ORIGINAL

An arithmetic sequence has first term $a$ and common difference $d$ . Its $8$ th term is $31$ and the sum of its first $20$ terms is $730$ . Find $a$ and $d$ , and find the least value of $n$ for which the nth term exceeds $100$ .

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 $-1$ .

Tier 1 · Easy

2 marks

ORIGINAL

Find the $8$ th term of the geometric sequence $3,6,12,\ldots$ .

Tier 2 · Standard

3 marks

ORIGINAL

Find the exact sum to infinity of $12-3+0.75-\cdots$ .

Tier 3 · Hard

5 marks

ORIGINAL

A geometric series has first term $800$ and common ratio $0.9$ . Find the least value of $n$ for which the sum of the first $n$ terms exceeds $7000$ .

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.

Tier 1 · Easy

2 marks

ORIGINAL

Mina saves $\pounds 25$ in the first month and increases the amount saved by $\pounds 5$ each month. Find the total she saves in the first $6$ months.

Tier 2 · Standard

4 marks

ORIGINAL

A ball is dropped from a height of $12$ m. After each impact it rebounds to $65\%$ of its previous maximum height. Find the total vertical distance travelled before it comes to rest, giving your answer to $3$ significant figures.

Tier 3 · Hard

6 marks

ORIGINAL

An account initially contains $\pounds 5000$ . At the end of each year, after interest of $3\%$ has been added, $\pounds 400$ is withdrawn. The model is $B_0=5000$ and $B_{n+1}=1.03B_n-400$ . Derive a formula for $B_n$ and find the first value of $n$ for which the model predicts a negative balance. State one limitation of the model.