FP1: SeriesEdexcel International A Level Further Maths: Topic test

1. 1
   A stack of logs is built in rows. Counting from the top, row $r$ contains $r$ logs, so the top row has $1$ log, the next row has $2$ logs, and so on.

   (a)

   Find the total number of logs in a stack with $15$ rows.

   [1 mark]

   • A$120$
   • B$240$
   • C$136$
   • D$105$

   (b)

   A second stack built in the same way contains $276$ logs in total. How many rows does it have?

   [1 mark]

   • A$24$
   • B$22$
   • C$23$
   • D$138$

   (c)

   Find the total number of logs in rows $11$ to $20$ inclusive.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The sum of the first $n$ square numbers, $1^2+2^2+\dots+n^2$, is denoted by $T_n$.

   (a)

   Find $T_{10}$.

   [1 mark]

   • A$3025$
   • B$506$
   • C$770$
   • D$385$

   (b)

   Which expression is equal to $\sum_{r=1}^{n}\left(r^2+r\right)$?

   [1 mark]

   • A$\frac{n(n+1)(n+2)}{6}$
   • B$\frac{n(n+1)(n+2)}{3}$
   • C$\frac{n(n+1)(2n+1)}{6}$
   • D$\frac{n^2(n+1)^2(2n+1)}{12}$

   (c)

   Find the value of $\sum_{r=11}^{20}r^2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence has $r$th term $u_r=(r+1)(r+3)$.

   (a)

   Show that $\sum_{r=1}^{n}u_r=\frac{n(2n^2+15n+31)}{6}$.

   [3 marks]

   (b)

   Hence find $\sum_{r=11}^{20}u_r$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $S_n=\sum_{r=1}^{n}(2r-1)^2$, the sum of the squares of the first $n$ odd numbers.

   (a)

   Show that $S_n=\frac{n(2n-1)(2n+1)}{3}$.

   [6 marks]

   (b)

   (i) Hence find the value of $21^2+23^2+25^2+\dots+59^2$. (ii) Find the least $n$ for which $S_n>10\,000$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A child saves money over several days. On day $r$ she puts $3r$ pence into one pot and $r^2$ pence into a second pot.

   (a)

   How many pence are in the first pot after $30$ days?

   [1 mark]

   • A$465$
   • B$2790$
   • C$1395$
   • D$1305$

   (b)

   How many pence are in the second pot after $12$ days?

   [1 mark]

   • A$650$
   • B$6084$
   • C$325$
   • D$78$

   (c)

   Find the total number of pence in the two pots together after $20$ days.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A sequence has $r$th term $a_r=2r+1$ and a second sequence has $r$th term $b_r=r^2+1$.

   (a)

   Which expression is equal to $\sum_{r=1}^{n}a_r$?

   [1 mark]

   • A$n(n+1)$
   • B$n(n+2)$
   • C$(n+1)^2$
   • D$n^2+n+1$

   (b)

   Which expression is equal to $\sum_{r=1}^{n}b_r$?

   [1 mark]

   • A$\frac{n(n+1)(2n+1)}{6}+1$
   • B$\frac{n(n+1)(2n+1)}{6}$
   • C$\frac{n(2n^2+3n+1)}{6}+n^2$
   • D$\frac{n(2n^2+3n+7)}{6}$

   (c)

   Find the value of $\sum_{r=1}^{10}\left(b_r-a_r\right)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A sequence has $r$th term $u_r=r(3r-1)$.

   (a)

   Show that $\sum_{r=1}^{n}u_r=n^2(n+1)$.

   [3 marks]

   (b)

   Find, in terms of $n$, a simplified expression for $\sum_{r=n+1}^{2n}u_r$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A sequence has $r$th term $v_r=(r-1)(r-2)$.

   (a)

   Show that $\sum_{r=1}^{n}v_r=\frac{n(n-1)(n-2)}{3}$.

   [6 marks]

   (b)

   (i) Hence find $\sum_{r=4}^{12}v_r$. (ii) Verify that $\sum_{r=1}^{n}v_r=2280$ when $n=20$, and explain why this is the only positive integer value of $n$ for which the sum is $2280$.

   [6 marks]

   Total for question 8: 12 marks

End of questions