CAIE A-Level Mathematics 1.6.3 Progression Formulae

Practise using nth-term and sum formulae for arithmetic and geometric progressions and forming linked equations when terms or partial sums create a second progression.

[Maximum number: 5]

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).

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1.6.3—Progression formulae question 1

Question (a)

Question (b)