Pure: Sequences and seriesEdexcel A-Level Maths: Topic test

1. 1
   The expression $(3-2x)^4$ is expanded in ascending powers of $x$.

   (a)

   What is the coefficient of $x^2$?

   [1 mark]

   • A$-108$
   • B$216$
   • C$54$
   • D$72$

   (b)

   What is the coefficient of $x^3$?

   [1 mark]

   • A$-96$
   • B$96$
   • C$-32$
   • D$-288$

   (c)

   Find the coefficient of $x^2$ in the expansion of $(1+x)(3-2x)^4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x)=(1+2x)^{-3}$.

   (a)

   What is the coefficient of $x$ in the binomial expansion of $g(x)$ in ascending powers of $x$?

   [1 mark]

   • A$6$
   • B$-3$
   • C$-6$
   • D$-12$

   (b)

   What is the coefficient of $x^2$ in the expansion of $g(x)$?

   [1 mark]

   • A$12$
   • B$6$
   • C$-24$
   • D$24$

   (c)

   Use the first three terms of the expansion of $g(x)$ with a suitable value of $x$ to estimate the value of $\dfrac{1}{1.02^3}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence is defined by $u_1=2$ and $u_{n+1}=ku_n+3$ for $n\geqslant1$, where $k$ is a constant.

   (a)

   Given that $u_3=17$, find the possible values of $k$.

   [3 marks]

   (b)

   Given that $k>0$, find the value of $\sum_{r=1}^{4}u_r$, and state, with a reason, whether the sequence is increasing, decreasing or periodic.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A runner follows a 20-week training plan, starting with a weekly distance of 12 km in week 1. Under Model P, each week's distance is 1.5 km more than the previous week's. Under Model Q, each week's distance is 8% more than the previous week's.

   (a)

   Using Model P, find the distance in week 20, the total distance over the 20 weeks, and the first week in which the weekly distance exceeds 30 km.

   [6 marks]

   (b)

   Using Model Q, find the first week in which the weekly distance exceeds 30 km, and the total distance over the 20 weeks. State which model gives the greater total.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   An arithmetic sequence has fourth term 23 and eleventh term 58.

   (a)

   What is the common difference?

   [1 mark]

   • A$35$
   • B$\frac{35}{8}$
   • C$\frac{35}{11}$
   • D$5$

   (b)

   What is the first term?

   [1 mark]

   • A$38$
   • B$18$
   • C$8$
   • D$3$

   (c)

   Find the sum of the first 20 terms.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A geometric series has first term $20$ and common ratio $-\frac12$.

   (a)

   What is the fifth term?

   [1 mark]

   • A$\frac54$
   • B$-\frac54$
   • C$-\frac58$
   • D$-\frac52$

   (b)

   What is the sum to infinity?

   [1 mark]

   • A$40$
   • B$\frac{40}{3}$
   • C$-\frac{40}{3}$
   • DIt does not exist, because $r$ is negative.

   (c)

   Find the sum of the first 6 terms.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The function $h$ is defined by $h(x)=(8-6x)^{\frac13}$.

   (a)

   Find the binomial expansion of $h(x)$ in ascending powers of $x$, up to and including the term in $x^2$, giving each coefficient in its simplest form.

   [3 marks]

   (b)

   State the range of values of $x$ for which the expansion is valid, and use $x=0.1$ to estimate $\sqrt[3]{7.4}$ to 3 decimal places.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A ball is dropped from a height of 2 m onto a horizontal floor. After each bounce it rises to 70% of the height from which it last fell.

   (a)

   Find the height to which the ball rises after the 4th bounce. Find also the number of the first bounce after which the ball rises to a height of less than 1 cm.

   [6 marks]

   (b)

   Find the total distance travelled by the ball before it comes to rest, according to the model.

   [6 marks]

   Total for question 8: 12 marks

End of questions