CAIE A-Level Mathematics 1.6 Series Question Bank

Practise binomial expansions, arithmetic and geometric progressions, finite sums and convergence decisions in Paper 1 calculations.

Exam points

select the relevant binomial term to find coefficients, constants or terms independent of x
use u_n, S_n and S∞ formulae after finding the common difference or ratio
check -1 < r < 1 before applying a geometric sum to infinity

Covered syllabus points

1.6.1—Binomial expansion 1.6.2—Progressions 1.6.3—Progression formulae 1.6.4—Geometric convergence

[Maximum number: 6]

(a)

Give the complete expansion of $\left(x+\frac{2}{x}\right)^{5}$ .

[ 2 ]

$x^{5}+10 x^{3}+40 x+\frac{80}{x}+\frac{80}{x^{3}}+\frac{32}{x^{5}}$ or $x^{5}+10 x^{3}+40 x+80 x^{-1}+80 x^{-3}+32 x^{-5}$

B2, 1, 0

B2, all terms correct, B1 5 terms correct. Terms must be simplified. Lists of terms allowed.

(b)

In the expansion of $\left(a+b x^{2}\right)\left(x+\frac{2}{x}\right)^{5}$ , the coefficient of x is zero and the coefficient of $\frac{1}{x}$ is 80 . Find the values of the constants a and b.

[ 4 ]

their $40 \times a+\left(\right.$ their coefficient of $\left.x^{-1}\right) \times b=0$

M1

Coefficients of a and b must be non-zero, allow x 's so
long as they are dealt with correctly.

(their coefficient of $\left.x^{-1}\right) \times a+\left(\right.$ their coefficient of $\left.x^{-3}\right) \times b=80$

M1

Coefficients of a and b must be non-zero, allow x 's as
long as they are dealt with correctly.

$a=2 \quad b=-1$

A1 A1

Dependent on both M marks, may be seen without working.

[Maximum number: 7]

A tool for putting fence posts into the ground is called a 'post-rammer'. The distances in millimetres that the post sinks into the ground on each impact of the post-rammer follow a geometric progression. The first three impacts cause the post to sink into the ground by $50 \mathrm{~mm}, 40 \mathrm{~mm}$ and 32 mm respectively.

(a)

Verify that the 9th impact is the first in which the post sinks less than 10 mm into the ground.

[ 3 ]

r=0.8

B1

SOI

$50 \times(\text { their } 0.8)^{7}=10.5$

M1

Evaluate $8{ }^{\text {th }}$ or $9{ }^{\text {th }}$ term in GP.

$50 \times(\text { their } 0.8)^{8}=8.39$ . Hence 9th impact required

A1

AG Two terms correct + conclusion (mention of $9^{\text {th }}$
impact or $u_{9}$ somewhere in the solution).
Statement that one is <10 (and the other >10) is
insufficient unless it mentions $9^{\text {th }}$ impact or $u_{9}$ .

Alternative method for final two marks: Logarithm method

$\begin{aligned} 50 \times(\text { their } 0.8)^{n}<10 \Rightarrow(\text { their } 0.8)^{n}<0.5 n \log (\text { their } 0.8)<\log 0.5 n>\frac{\log 0.5}{\log (\text { their } 0.8)} \Rightarrow[n>] 7.2 \end{aligned}$

M1

n=8 hence $9^{\text {th }}$ impact required

A1

AG Need conclusion that mentions $9^{\text {th }}$ impact or $u_{9}$ .

(b)

Find, to the nearest millimetre, the total depth of the post in the ground after 20 impacts.

[ 2 ]

$\frac{50\left(1-(\text { their } 0.8)^{20}\right)}{1-\text { their } 0.8}$

M1

OE Use of formula with their r SOI.

=247

A1

Must be to the nearest mm (not 247.1).

(c)

Find the greatest total depth in the ground which could theoretically be achieved.

[ 2 ]

$\frac{50}{1-\text { their } 0.8}$

M1

Use of sum to infinity formula with their r SOI.
Substituting a value of n into the sum formula M0.

=250

A1

All question bank results loaded

Covered syllabus points

1.6 Series question 1

Question (a)

Question (b)

1.6 Series question 2

Question (a)

Question (b)

Question (c)