Edexcel A-Level Maths: Sequences & Series

When Priya starts a new job her annual salary is £24 000. Her contract guarantees that at the start of each subsequent year her salary will increase by 4% of its value in the previous year. Her salary in the first year is $u_1 = £24\,000$u1​=£24000, in the second year $u_2$u2​, and so on.

(a) Show that the salaries $u_1, u_2, u_3, \dots$u1​,u2​,u3​,… form a geometric sequence, and write down the first term and the common ratio. (3 marks)

(b) Find Priya's salary in her 10th year, and the total amount she earns over her first 10 years. Give each answer to the nearest pound. (5 marks)

(c) Find the smallest number of complete years for which Priya's total earnings first exceed £400 000. (4 marks)

An arithmetic sequence has first term $a$a and common difference $d$d. The 5th term of the sequence is 17 and the 12th term is 38.

(a) Use this information to write down two equations in $a$a and $d$d. (3 marks)

(b) Solve your equations to find the value of $a$a and the value of $d$d. (3 marks)

(c) The sum of the first $n$n terms of the sequence is $S_n$Sn​. Find the smallest value of $n$n for which $S_n > 1000$Sn​>1000. (4 marks)

(a) Find the binomial expansion of $(1 + 4x)^{1/2}$(1+4x)1/2 in ascending powers of $x$x, up to and including the term in $x^3$x3. Give each coefficient as an integer. (5 marks)

(b) State the range of values of $x$x for which the expansion is valid. (1 mark)

(c) By substituting a suitable value of $x$x into your expansion, find an approximation for $\sqrt{1.04}$1.04​, giving your answer to 6 decimal places. (2 marks)

(a) Show that $\displaystyle\sum_{r=1}^{n}(3r - 2) = \frac{n(3n - 1)}{2}$r=1∑n​(3r−2)=2n(3n−1)​. (4 marks)

(b) Hence find the exact value of $\displaystyle\sum_{r=1}^{20}(3r - 2)$r=1∑20​(3r−2). (2 marks)

A geometric series has first term 18 and second term 12.

(a) Find the common ratio $r$r, and explain why the sum to infinity of this series exists. (2 marks)

(b) Find the sum to infinity of the series. (3 marks)

A sequence is defined by the recurrence relation

$u_{n+1} = \frac{1}{1 - u_n}, \qquad u_1 = 2.$un+1​=1−un​1​,u1​=2.

(a) Find the values of $u_2$u2​, $u_3$u3​ and $u_4$u4​, and state the period of the sequence. (2 marks)

(b) Hence find the exact value of $\displaystyle\sum_{n=1}^{30} u_n$n=1∑30​un​. (2 marks)