P2: Sequences and seriesEdexcel International A Level Maths: Topic test

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

   (a)

   Find the value of $u_3$.

   [1 mark]

   • A$11$
   • B$15$
   • C$19$
   • D$3$

   (b)

   Which statement about this sequence is true?

   [1 mark]

   • AThe sequence is decreasing.
   • BThe sequence is increasing.
   • CThe sequence is periodic.
   • DEvery term of the sequence is even.

   (c)

   Find the value of $\displaystyle\sum_{r=1}^{5}u_r$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A runner trains every day. On day 1 she runs $3$ km and on each later day she runs $0.5$ km further than on the previous day.

   (a)

   How far does she run on day $15$?

   [1 mark]

   • A$10.5$ km
   • B$10$ km
   • C$7.5$ km
   • D$18$ km

   (b)

   What is the total distance she runs in the first $15$ days?

   [1 mark]

   • A$101.25$ km
   • B$195$ km
   • C$97.5$ km
   • D$112.5$ km

   (c)

   Find the total distance she runs on days $10$ to $20$ inclusive.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A geometric series has positive terms. Its second term is $12$ and its sum to infinity is $54$.

   (a)

   Show that the common ratio $r$ satisfies $9r^2-9r+2=0$.

   [3 marks]

   (b)

   Given that $r>\frac12$, find the smallest value of $n$ for which the sum of the first $n$ terms of the series exceeds $53.9$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The coefficients of $x$, $x^2$ and $x^3$ in the expansion of $(1+x)^n$, where $n\ge3$ is an integer, are three consecutive terms of an arithmetic sequence.

   (a)

   Show that $n^2-9n+14=0$ and hence find the value of $n$.

   [6 marks]

   (b)

   The coefficients of $x$, $x^2$ and $x^3$ for the value of $n$ found in part (a) are the first three terms of an arithmetic series. Find the sum of the first $20$ terms of this series, and the smallest number of terms for which the sum exceeds $5000$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider the expansion of $(3+2x)^5$ in ascending powers of $x$.

   (a)

   What is the coefficient of $x^3$?

   [1 mark]

   • A$720$
   • B$1080$
   • C$180$
   • D$90$

   (b)

   Use the first three terms of this expansion, with a suitable value of $x$, to estimate $3.02^5$.

   [1 mark]

   • A$251.1$
   • B$251.208$
   • C$259.632$
   • D$261.9$

   (c)

   Find the coefficient of $x^2$ in the expansion of $(1-x)(3+2x)^5$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A sequence is defined by $u_1=1$ and $u_{n+1}=-\dfrac{1}{u_n+1}$ for $n\ge1$.

   (a)

   Find the value of $u_3$.

   [1 mark]

   • A$2$
   • B$-\frac12$
   • C$-2$
   • D$-\frac32$

   (b)

   Find the value of $u_{50}$.

   [1 mark]

   • A$1$
   • B$-2$
   • C$\frac12$
   • D$-\frac12$

   (c)

   Find the value of $\displaystyle\sum_{r=1}^{100}u_r$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The sum of the first $n$ terms of an arithmetic series is $S_n=2n^2+3n$.

   (a)

   Find the first term and the $n$th term of the series.

   [3 marks]

   (b)

   The $k$th term is the first term of the series to exceed $150$. Find $k$ and $S_k$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A charity receives £8000 in donations in its first year. In each later year, donations are $10\%$ lower than in the previous year.

   (a)

   Find the donations in year $6$ to the nearest pound, the total donations over the first $10$ years to the nearest pound, and the greatest total the charity could ever receive.

   [6 marks]

   (b)

   The charity needs total donations of at least £70000. Find the smallest number of years this takes. Explain why a total of £81000 can never be reached.

   [6 marks]

   Total for question 8: 12 marks

End of questions