CAIE A-Level Mathematics AS 1.6 Series Questions

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: 4]

Find the coefficient of $x^{2}$ in the expansion of

(2-5 x)(1+3 x)^{10}

Correct second term 30 x in expansion of $(1+3 x)^{10}$
WWW, may be implied later.

Correct third term $+405 x^{2}$
Marking guidance:

Ignore subsequent terms, may be implied later.

Multiply (2-5x) by their $30 x+405 x^{2}$ to obtain two $x^{2}$ terms only
Expect $-150 x^{2}, 810 x^{2}$ .

Coefficient is 660
Must be clearly identified.
Allow final answer $660 x^{2}$ .

[Maximum number: 7]

The first term of an arithmetic progression is 1.5 and the sum of the first ten terms is 127.5 .

(a)

Find the common difference.

[ 2 ]

(b)

Find the sum of all the terms of the arithmetic progression whose values are between 25 and 100 .

[ 5 ]

Attempt to find either the first term or the last term in the set by considering
1.5+2.5(n-1)>25 or 1.5+2.5(n-1)<100 or equivalent equations
Using their d.
May be implied by correct answers.

State or imply that 11th term or 26.5 is the first in the set
State or imply that 40th term or 99 is the last in the set
Either use $S_{40}-S_{10}$

Or use $\frac{1}{2} n(a+l)$ with correct results for their d Or use $\frac{1}{2} n[2 a+(n-1) d]$ with correct results for their d
Their 40 and 10 from correct working with their
d.
Correct values 30, 26.5 and 99 respectively.
Correct values 30, 26.5 and 2.5 respectively.

Obtain 1882.5
OE

[Maximum number: 8]

The geometric progression $a_{1}, a_{2}, a_{3}, \ldots$ has first term 2 and common ratio r where r>0. It is given that $\frac{9}{2} a_{5}+7 a_{3}=8$ .

(a)

Find the value of r.

[ 3 ]

Substitute to obtain equation $9 r^{4}+14 r^{2}-8=0$
OE

Attempt solution of quadratic equation in $r^{2}$ to obtain at least one value of r or $r^{2}$
Expect $\left(9 r^{2}-4\right)\left(r^{2}+2\right)$ .

$r=\frac{2}{3}$ only
SC B1 answer without working.

(b)

Find the sum of the first 20 terms of the geometric progression. Give your answer correct to 4 significant figures.

Figure for Question (b) — CAIE A-Level Mathematics AS

[ 2 ]

Substitute a=2 and their r in correct formula and attempt to evaluate
Expect $\frac{2\left(1-\left(\frac{2}{3}\right)^{20}\right)}{\left(1-\frac{2}{3}\right)}$ or $\frac{2\left(\left(\frac{2}{3}\right)^{20}-1\right)}{\left(\frac{2}{3}-1\right)}$ .

5.998
AWRT and no other value.

(c)

Find the sum to infinity of the progression $a_{2}, a_{5}, a_{8}, \ldots$ .

[ 3 ]

Identify $a_{2}=\frac{4}{3}$ and common ratio as $\frac{8}{27}$ .

B1 FT

Following their r provided |r|<1.
May be implied in the sum to infinity.

Marking guidance:

Allow $\left(\frac{2}{3}\right)^{3}$ .

Substitute their new a and r in correct formula for sum to infinity and evaluate
|r|<1 otherwise M0.

$\frac{36}{19}$
OE
Accept 1.89 or better from 1.894736.....

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)

Question (c)