Question difficulty level:
Basic
Moderate
Challenging
Difficult
Genius :)
Basic
Moderate
Challenging
Difficult
Genius :)
Paper 1 – No Calculator
[Maximum mark: 5]
The first three terms of an arithmetic sequence are 2, 5, and 8.
(a) Write down d, the common difference of the sequence.
[1]
(b) Find the 11th term of the sequence.
[2]
(c) Find the sum of the first 11 terms of the sequence.
[2]
(a)
d = 5 - 2
{\boldsymbol{\color{Purple} d = 3}}
(b)
Using the nth term of an arithmetic sequence formula, we get that:
u_{11} = 2 + (11-1)(3) = 2 + 30
\boldsymbol{{\color{Purple} u_{11} = 32}}
(c)
Using the sum of n terms of an arithmetic sequence formula (the second version in the Formula Booklet), we get that:
S_{11} = \frac{11}{2}\left ( 2 + 32 \right ) = 11 \times 17
\boldsymbol{{\color{Purple} S_{11} = 187}}
.
Paper 1 – No Calculator
[Maximum mark: 5]
The third term of an arithmetic sequence is 8, and the fifth term of the sequence is 14.
(a) Find
(i) the common difference of the sequence.
(ii) the first term of the sequence.
[3]
The nth term of the sequence is 35.
(b) Find the value of n.
[2]
.(a) (i)
d = \frac{u_{5} - u_{3}}{2} = \frac{14-8}{2}
\boldsymbol{{\color{Purple} d = 3}}
(a) (ii)
u_{1} = u_{3} - 2d = 8 - 2(3)
\boldsymbol{{\color{Purple} u_{1} = 2}}
(b)
Using the nth term of an arithmetic sequence formula:
35 = 2 + (n - 1)(3)
35 = 3n - 1
\boldsymbol{{\color{Purple} n = 12}}
.
Paper 2 – Calculator
[Maximum mark: 6]
The sum to infinity of a geometric sequence is 100 and the sum of the first five terms of the sequence is 67.232.
Find the common ratio of the sequence.
Using the sum of an infinite geometric sequence formula and we can write that:
100 = \frac{u_{1}}{1 - r}
Using the sum of n terms of a finite geometric sequence we can write that:
67.232 = \frac{u_{1}(1 - r^{5})}{1 - r}
Rearranging the first equation we get that:
u_{1} = 100(1 - r)
Substituting 100(1 - r) for u_{1} in the second equation we can write that:
67.232 = \frac{(100(1 - r))(1 - r^{5})}{1 - r}
Simplifying gives us:
67.232 = 100(1 - r^{5})
Solving for r either by graphing or using the Solver function on the calculator, we get that:
\boldsymbol{{\color{Purple} r = 0.8}}
.
Paper 2 – Calculator
[Maximum mark: 6]
The first term of an arithmetic sequence is -86 and the common difference of the sequence is 4.
The nth term of the sequence is denoted by u_{n}.
(a) Find the smallest possible value of n for which u_{n} > 0.
[3]
The sum of the first n terms of the sequence is denoted by S_{n}.
(b) Find the minimum value of S_{n}.
[3]
(a)
Using the nth term of an arithmetic sequence formula we get:
u_{n} = -86 + (n-1) \times 4
Substituting zero for u_{n} and solving for n we get:
0 = -86 + (n-1) \times 4
n = 22.5
Since n must be a whole number, the final answer is:
\boldsymbol{{\color{Purple} n = 23}}
(b)
Using the sum of the first n terms of a arithmetic sequence formula we get:
S_{n} = \frac{n}{2}\left ( 2(-86) + (n-1) \times 4 \right )
Finding the minimum value of this expression, for example, by graphing on the calculator, we get that the minimum value of S_{n} is:
\boldsymbol{{\color{Purple} -968}}
.
Paper 2 – Calculator
[Maximum mark: 8]
Consider the first three terms of the following sequences:
Sequence 1: 9; 3; 1
Sequence 2: 2; 4; 7
Sequence 3: \frac{1}{2}; \frac{1}{3}; \frac{1}{4}
Sequence 4: 6; 2; -2
(a) State which sequence is
(i) arithmetic.
(ii) geometric.
[2]
(b) For the arithmetic sequence
(i) write down the common difference.
(ii) find the 15th term.
[3]
(c) For the geometric sequence
(i) write down the common ratio.
(ii) find the exact value of the sum of the first seven terms.
[3]
(a) (i)
\boldsymbol{{\color{Purple} \textbf{Sequence 4}}}
(a) (ii)
\boldsymbol{{\color{Purple} \textbf{Sequence 1}}}
(b) (i)
d = 2 - 6
\boldsymbol{{\color{Purple} d = -4}}
(b) (ii)
Using the nthe term of an arithmetic sequence formula:
u_{15} = 6 + (15-1)(-4)
\boldsymbol{{\color{Purple} u_{15} = -50}}
(c) (i)
\boldsymbol{{\color{Purple} r = \frac{1}{3}}}
(c) (ii)
Using the sum of nterms of a finite geometric sequence formula:
S_{7} = \frac{9\left ( 1 - \left ( \frac{1}{3}\right )^{7} \right )}{1 - \frac{1}{3}}
\boldsymbol{{\color{Purple} S_{7} = \frac{1093}{81}}}
.
Paper 1 – No Calculator
[Maximum mark: 14]
The first three terms of a sequence are 2\textup{log}_{a}x\: ,\: k\textup{log}_{a}x\: ,\: \textup{log}_{a}x where x\: ,\: a\: ,\: k\: \epsilon \: \mathbb{R} and x > 1\: ,\: a > 1\: ,\: k \neq 0.
(a) Assume that the sequence is arithmetic.
(i) Find the value of k.
(ii) Given that the common difference is m\textup{log}_{a}x where m\: \epsilon \: \mathbb{R}, write down the value of m.
(iii) Given that a = \sqrt{3} and S_{8} = 4, find the value of x.
[7]
(b) Now assume that the sequence is geometric.
(i) Given that k < 0, find the value of k.
(ii) Hence find an expression for a in terms of x, given that S_{\infty } = \frac{5 \sqrt{2}}{1 + \sqrt{2}}.
[7]
(a) (i)
d = u_{2} - u_{1} = u_{3} - u_{2}
k\textup{log}_{a}x - 2\textup{log}_{a}x = \textup{log}_{a}x - k\textup{log}_{a}x
k - 2 = 1 - k, therefore:
\boldsymbol{{\color{Purple} k = \frac{3}{2}}}
(a) (ii)
d = u_{2} - u_{1} = \frac{3}{2}\textup{log}_{a}x - 2\textup{log}_{a}x = -\frac{1}{2}\textup{log}_{a}x
\boldsymbol{{\color{Purple} m = -\frac{1}{2}}}
(a) (iii)
Substituting into the sum of the first n terms of an arithmetic sequence formula:
4 = \frac{8}{2}\left ( 2 \times 2\textup{log}_{\sqrt{3}}\, x + (8-1)\left ( -\frac{1}{2}\textup{log}_{\sqrt{3}}\, x \right ) \right )
Simplifying and solving for x:
1 = 4\textup{log}_{\sqrt{3}}\, x - 3.5\textup{log}_{\sqrt{3}}\, x
\textup{log}_{\sqrt{3}}\, x = 2
x = (\sqrt{3})^{2}, so:
\boldsymbol{{\color{Purple} x = 3}}
(b) (i)
r = \frac{u_{2}}{u_{1}} = \frac{u_{3}}{u_{2}}
\frac{k\textup{log}_{a}x}{2\textup{log}_{a}x} = \frac{\textup{log}_{a}x}{k\textup{log}_{a}x}
k^{2} = 2 \Rightarrow k = \pm \sqrt{2}
\boldsymbol{{\color{Purple} k = -\sqrt{2}}}
(b) (ii)
r = \frac{\textup{log}_{a}x}{-\sqrt{2}\textup{log}_{a}x} = -\frac{1}{\sqrt{2}}
Using the sum of an infinite geometric sequence formula:
\frac{5 \sqrt{2}}{1 + \sqrt{2}} = \frac{2\textup{log}_{a}x}{1 - \left ( -\frac{1}{\sqrt{2}} \right )}
Simplifying and multiplying the fraction on the right-hand side by \frac{\sqrt{2}}{\sqrt{2}}:
\frac{5 \sqrt{2}}{1 + \sqrt{2}} = \frac{2 \sqrt{2}\textup{log}_{a}x}{1 + \sqrt{2}}
\textup{log}_{a}x = \frac{5}{2}\Rightarrow x = a^{\frac{5}{2}}
\boldsymbol{{\color{Purple} a = x^{\frac{2}{5}}}}
.