Modelling with sequences and seriesAQA A-Level Maths: Subtopic test

1. 1
   A company's profit in year $1$ is £$40\,000$ and it increases by £$5\,000$ each year.

   (a)

   Find the profit in year $10$.

   [1 mark]

   • A£$90\,000$
   • B£$80\,000$
   • C£$625\,000$
   • D£$85\,000$

   (b)

   Find the total profit over the first $10$ years.

   [1 mark]

   • A£$625\,000$
   • B£$850\,000$
   • C£$650\,000$
   • D£$125\,000$

   (c)

   Find the first year in which the profit exceeds £$200\,000$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A ball is dropped from a height of $2$ m. After each bounce it rises to $70\%$ of the height from which it last fell.

   (a)

   Find the height reached after the third bounce.

   [1 mark]

   • A$0.98$ m
   • B$0.48$ m
   • C$0.686$ m
   • D$1.4$ m

   (b)

   Which expression gives the height reached after the $n$th bounce?

   [1 mark]

   • A$2(0.7)^{n-1}$
   • B$2(0.7)^n$
   • C$2(0.7n)$
   • D$0.7(2)^n$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A person invests £$1000$ at the start of year $1$ and a further £$1000$ at the start of each following year. Interest of $4\%$ per year is added at the end of each year.

   (a)

   Find the total value of the investment at the end of year $5$, to the nearest pound.

   [3 marks]

   (b)

   Find the number of years after which the total value first exceeds £$20\,000$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A loan of £$11\,800$ is repaid by monthly payments. The first payment is £$400$ and each payment is £$20$ more than the one before.

   (a)

   Show that the number of months, $n$, needed to repay the loan satisfies $n^2+39n-1180=0$, and hence find $n$.

   [6 marks]

   (b)

   The loan is repaid in exactly $20$ months.
   (i) Find the final payment.
   (ii) Find the total paid in the last $5$ months.
   (iii) Find the mean monthly payment.

   [6 marks]

   Total for question 4: 12 marks

End of questions