Sequences and seriesAQA A-Level Maths: Topic test

1. 1
   The terms of an arithmetic sequence satisfy $u_n=65-4n$ for $n\ge1$.

   (a)

   What is the common difference of the sequence?

   [1 mark]

   • A$-4$
   • B$4$
   • C$61$
   • D$65$

   (b)

   How many terms of the sequence are positive?

   [1 mark]

   • A$15$
   • B$17$
   • C$16$
   • D$65$

   (c)

   Find $\sum_{r=1}^{n}u_r$ for the positive terms, that is the sum of all the positive terms of the sequence.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A geometric series has first term $54$ and third term $24$, and all of its terms are positive.

   (a)

   What is the common ratio?

   [1 mark]

   • A$\dfrac49$
   • B$\dfrac23$
   • C$-\dfrac23$
   • D$\dfrac94$

   (b)

   What is the sum to infinity?

   [1 mark]

   • A$108$
   • B$81$
   • C$54$
   • D$162$

   (c)

   Find the smallest number of terms for which the sum exceeds $160$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=(8-3x)^{\frac13}$.

   (a)

   Find the first three terms in the binomial expansion of $\mathrm{f}(x)$ in ascending powers of $x$.

   [3 marks]

   (b)

   State the range of values of $x$ for which the expansion is valid. Use the first three terms with $x=0.04$ to estimate $\sqrt[3]{7.88}$, giving your answer to $5$ decimal places.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sequence is defined by $u_1=k$ and $u_{n+1}=3-u_n$ for $n\ge1$, where $k$ is a constant.

   (a)

   (i) Find $u_2$ and $u_3$ in terms of $k$, and state what type of sequence this is. (ii) Find $\sum_{r=1}^{40}u_r$. (iii) Find $\sum_{r=1}^{101}u_r$ in terms of $k$.

   [6 marks]

   (b)

   Given that $\sum_{r=1}^{101}u_r=136$, (i) find $k$ and the value of $u_{2025}$. (ii) A student says the sequence is increasing because $u_2>u_1$. Explain why this is wrong. (iii) Find the least even $n$ for which $\sum_{r=1}^{n}u_r>1000$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A sequence is defined by $u_1=3$ and $u_{n+1}=u_n^2-4u_n+5$ for $n\ge1$.

   (a)

   What is the value of $u_3$?

   [1 mark]

   • A$5$
   • B$2$
   • C$3$
   • D$1$

   (b)

   What is the value of $u_{100}$?

   [1 mark]

   • A$1$
   • B$2$
   • C$3$
   • D$5$

   (c)

   Find $\sum_{r=1}^{50}u_r$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A runner trains for $30$ days. On day $1$ she runs $2$ km and on each following day she runs $0.25$ km further than on the previous day.

   (a)

   How far does she run on day $21$?

   [1 mark]

   • A$5.25$ km
   • B$7.25$ km
   • C$7$ km
   • D$8$ km

   (b)

   What is the total distance she runs in the $30$ days?

   [1 mark]

   • A$168.75$ km
   • B$337.5$ km
   • C$217.5$ km
   • D$11.25$ km

   (c)

   The runner's target is to run a total of more than $100$ km. Find the first day on which her total distance exceeds $100$ km.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A pendulum swings through an arc of $80$ cm on its first swing. Each later swing is $92\%$ of the length of the swing before it.

   (a)

   Find the length of the $12$th swing, to $3$ significant figures.

   [3 marks]

   (b)

   Find the least number of swings for which the total distance travelled exceeds $900$ cm.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The binomial expansion of $(1+px)^n$, where $n$ is a rational number, begins $1+3x-\dfrac92x^2+\dots$

   (a)

   Find the values of $p$ and $n$, and the range of values of $x$ for which the expansion is valid.

   [6 marks]

   (b)

   Use the expansion to estimate $\sqrt{1.06}$ with $x=0.01$. Find the coefficient of $x^3$ and say what including the $x^3$ term does to the estimate. The true value is $1.029563\dots$

   [6 marks]

   Total for question 8: 12 marks

End of questions