• A Level Maths

Demo

• A Level
• Sequences

1. Write down the first 4 terms of the sequence with $n$th term given by $u_n = 2^n$.

$2,4,8,16$

2. The $n$th term of the sequence $$4,7,10,13, ...$$ is given by $u_n = an+b$. Find $u_n$.

$3n+1$

3. Suggest possible expressions for the $n$th term of the following sequences:

1. $3,9,27,81,243...$
2. $0,1,8,27,64...$

1. $3^n$
2. $(n-1)^3$

4. The $n$th term of a sequence is given by $$u_n = a + 3^{n-2}$$ Given that $u_3 = 11$, find $u_6$.

$89$

5. The $n$th term of a sequence is given by $$u_n = n(2n+k)$$ Given $u_6 = 2u_4-2$, prove that $$u_n - u_{n-1} = an+b$$ for all values of $n$, where $a$ and $b$ are constants to be found.

$4n+3$

6. The $n$th term of a sequence is given by $$u_n = k^n-3$$ Given that $u_1+u_2 = 0$, find two possible values for $u_5$.

$-246, 29$

7. Write down the first 4 terms of the following sequences:

1. $u_n = u_{n-1}+4$, $u_1 = 3$
2. $u_{n+1} = 2u_n+5$, $u_1 = -2$

1. $3,7,11,15$
2. $-2,1,7,19$

8. The sequence $$-4,-3,-1,3,11$$ can be defined by the recurrence relation $u_n = au_{n-1} + b$. Find $a$ and $b$.

$a=2,b=5$

9. For the following sequences, find expressions for $u_3$ in terms of $k$:

1. $u_n = 4u_{n-1}-k \quad u_1 = k$
2. $u_{n+1} = \dfrac{u_n}{k} \quad u_1 = 4$

1. $11k$
2. $\dfrac{4}{k^2}$

10. For the following sequences, find expressions for $u_3$ in terms of $k$:

1. $u_n = 2-ku_{n-1} \quad u_1 = -1$
2. $u_{n+1} = \sqrt[3]{61k^3+u_n^3} \quad u_1 = k\sqrt[3]{3}$

1. $2-2k-k^2$
2. $5k$

11. Given $$u_n = \frac{1}{2}(k+3u_{n-1}) \quad u_1 = 2 \quad u_3 = 7$$ find $u_4$.

$\dfrac{23}{2}$

12. For the following sequences, find $u_1$:

1. $u_n = \dfrac{1}{2}\sqrt{u_{n-1}} \quad u_3 = 1$
2. $u_{n+1} = 0.2(1-u_n) \quad u_3 = -0.2$

1. $64$
2. $-9$

13. For the following sequences, find $u_1$ and $u_4$:

1. $u_n = 3u_{n-1}-2 \quad u_3 = 10$
2. $u_{n+1} = \dfrac{3u_n}{4}+2 \quad u_3 = 5$

1. $u_1 = 2$, $u_4 = 28$
2. $u_1 = \dfrac{8}{3}$, $u_4 = \dfrac{23}{4}$

14. Given $$u_{n+1} = u_n + c \quad u_1 = 2 \quad u_5 = 30$$ find an expression for $u_n$ in terms of $n$.

$7n-5$

15. Given $$u_n = 3(u_{n-1} - k) \quad u_1 = -4 \quad u_3 = 7u_2 +3$$ find the value of $u_4$.

$u_4 = 87$

16. A sequence is defined by $$u_n = ku_{n-1} + 2 \quad u_1 = 1.5$$ Find the possible values of $k$ if $u_3 = 12$.

$-\dfrac{10}{3}, 2$

17. The first term of an arithmetic sequence is $14$. The fourth term is $32$. What is the common difference?

$6$

18. For an arithmetic sequence, the 20th term is $14$ and the 40th term is $-6$. Find the 10th term.

$24$

19. Find the number of terms of these arithmetic sequences:

1. $90,88,..., 16,14$
2. $x,3x,5x,...,35x$

1. $39$
2. $18$

20. A geometric sequence has first term $4$ and third term $1$. Find two possible values for the 8th term.

$\pm\dfrac{1}{32}$

21. The first 3 terms of an arithmetic sequence are $-8,k^2,17k$. Find two possible values for $k$.

$\dfrac{1}{2}, 8$

22. An arithmetic sequence has first term $x^2$ and common difference $x$, where $x>0$. The fifth term is $41$. Find $x$ in the form $p+q\sqrt{5}$.

$-2+3\sqrt{5}$

23. The first three terms of a geometric sequence are $9,36,144$. State whether $383,616$ is a term in the sequence.

$383616 = 2^7\times 3^4\times 37^1$, not in the sequence.

24. The first 3 terms of a geometric sequence are given by $$8-x \quad 2x \quad x^2$$ where $x>0$. Find the value of the 16th term.

$131072$