1.04e Sequences: nth term and recurrence relations

A sequence is defined as follows. \(u_1 = a\), where \(a > 0\) To obtain \(u_{r+1}\)

• find the remainder when \(u_r\) is divided by 3,
• multiply the remainder by 5,
• the result is \(u_{r+1}\).

Find \(\sum_{r=2}^4 u_r\) in each of the following cases.

1. \(a = 5\)
2. \(a = 6\) [3]

The terms, \(u_n\), of a periodic sequence are defined by $$u_1 = 3 \quad \text{and} \quad u_{n+1} = \frac{-6}{u_n}$$

1. Find \(u_2\), \(u_3\) and \(u_4\) [2 marks]
2. State the period of the sequence. [1 mark]
3. Find the value of \(\sum_{n=1}^{101} u_n\) [2 marks]

Given \(u_1 = 1\), determine which one of the formulae below defines an increasing sequence for \(n \geq 1\) Circle your answer. [1 mark] \(u_{n+1} = 1 + \frac{1}{u_n}\) \quad \(u_n = 2 - 0.9^{n-1}\) \quad \(u_{n+1} = -1 + 0.5u_n\) \quad \(u_n = 0.9^{n-1}\)

A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket. When the rain stops, the bucket is one third full. Water continues to drip into the bucket from a puddle on the roof. In the first minute after the rain stops, 30 millilitres of water drips into the bucket. In each subsequent minute, the amount of water that drips into the bucket reduces by 2%. During the \(n\)th minute after the rain stops, the volume of water that drips into the bucket is \(W_n\) millilitres.

1. Find \(W_2\) [1 mark]
2. Explain why $$W_n = A \times 0.98^{n-1}$$ and state the value of \(A\). [2 marks]
3. Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre. [2 marks]
4. Assuming it does not start to rain again, find the maximum amount of water in the bucket. [3 marks]
5. After several hours the water has stopped dripping. Give two reasons why the amount of water in the bucket is not as much as the answer found in part (d). [2 marks]

The sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 0 \quad u_{n+1} = \frac{5}{6 - u_n}$$ Prove by induction that, for all integers \(n \geq 1\), $$u_n = \frac{5^n - 5}{5^n - 1}$$ [6 marks]

A financial institution models the repayment of a loan to a client in the following way.

• An amount, \(£C\), is loaned to the client at the start of the repayment period.
• The amount owed \(n\) years after the start of the repayment period is \(£L_n\), so that \(L_0 = C\).
• At the end of each year, interest of \(\alpha\%\) (\(\alpha > 0\)) of the amount owed at the start of that year is added to the amount owed.
• Immediately after interest has been added to the amount owed a repayment of \(£R\) is made by the client.
• Once \(L_n\) becomes negative the repayment is finished and the overpayment is refunded to the client.

1. Show that during the repayment period, \(L_{n+1} = aL_n + b\), giving \(a\) and \(b\) in terms of \(\alpha\) and \(R\). [2]
2. Find the solution of the recurrence relation \(L_{n+1} = aL_n + b\) with \(L_0 = C\), giving your solution in terms of \(a\), \(b\), \(C\) and \(n\). [5]
3. Deduce from parts (a) and (b) that, for the repayment scheme to terminate, \(R > \frac{\alpha C}{100}\). [2]

A client takes out a £30000 loan at 8% interest and agrees to repay £3000 at the end of each year.

1. 1. Use an algebraic method to find the number of years it will take for the loan to be repaid. [3]
   2. Taking into account the refund of overpayment, find the total amount that the client repays over the lifetime of the loan. [3]

A sequence \(u_1, u_2, u_3, ...\) is defined by \(u_n = 3n - 1\), for \(n \geq 1\).

1. Find the values of \(u_1, u_2, u_3\). [1]
2. Find $$\sum_{n=1}^{40} u_n$$ [3]

In this question you must show detailed reasoning. A sequence \(t_1, t_2, t_3 \ldots\) is defined by \(t_n = 5 - 2n\). Use an algebraic method to find the smallest value of \(N\) such that $$\sum_{n=1}^{\infty} 2^{t_n} - \sum_{n=1}^{N} 2^{t_n} < 10^{-8}$$ [8]

A sequence \(u_1, u_2, u_3 \ldots\) is defined by \(u_1 = 7\) and \(u_{n+1} = u_n + 4\) for \(n \geq 1\).

1. State what type of sequence this is. [1]
2. Find \(u_{17}\). [2]

A sequence of numbers \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 3$$ $$a_{n+1} = \frac{a_n - 3}{a_n - 2}, \quad n \in \mathbb{N}$$

1. Find \(\sum_{r=1}^{100} a_r\) [3]
2. Hence find \(\sum_{r=1}^{100} a_r + \sum_{r=1}^{99} a_r\) [1]

A sequence is defined by $$u_1 = 3$$ $$u_{n+1} = 2 - \frac{4}{u_n}, \quad n \geq 1$$ Find the exact values of

1. \(u_2\), \(u_3\) and \(u_4\) [3]
2. \(u_{61}\) [1]
3. \(\sum_{i=1}^{99} u_i\) [3]

A sequence is defined by $$u_1 = 600$$ $$u_{n+1} = pu_n + q$$ where \(p\) and \(q\) are constants. It is given that \(u_2 = 500\) and \(u_4 = 356\)

1. Find the two possible values of \(u_3\) [5 marks]
2. When \(u_n\) is a decreasing sequence, the limit of \(u_n\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\). [2 marks]

The sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_{n+1} = k - \frac{24}{u_n} \quad u_1 = 2$$ where \(k\) is an integer. Given that \(u_1 + 2u_2 + u_3 = 0\)

1. show that $$3k^2 - 58k + 240 = 0$$ [3]
2. Find the value of \(k\), giving a reason for your answer. [2]
3. Find the value of \(u_3\) [1]

A sequence of numbers \(a_1, a_2, a_3, \ldots\) is defined by $$a_{n+1} = \frac{k(a_n + 2)}{a_n}$$, \(n \in \mathbb{N}\) where \(k\) is a constant. Given that

• the sequence is a periodic sequence of order 3
• \(a_1 = 2\)

1. show that $$k^2 + k - 2 = 0$$ [3]
2. For this sequence explain why \(k \neq 1\) [1]
3. Find the value of $$\sum_{r=1}^{80} a_r$$ [3]

A sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 8 \quad \text{and} \quad u_{n+1} = u_n + 3.$$

1. Show that \(u_5 = 20\). [1]
2. The \(n\)th term of the sequence can be written in the form \(u_n = pn + q\). State the values of \(p\) and \(q\). [2]
3. State what type of sequence it is. [1]
4. Find the value of \(N\) such that \(\sum_{n=1}^{2N} u_n - \sum_{n=1}^{N} u_n = 1256\). [5]

A sequence of terms \(a_1, a_2, a_3, ...\) is defined by $$a_1 = 4$$ $$a_{n+1} = ka_n + 3$$ where \(k\) is a constant. Given that • \(\sum_{n=1}^{5} a_n = 12\) • all terms of the sequence are different find the value of \(k\) [4]

A sequence \(t_1, t_2, t_3, t_4, t_5, \ldots\) is given by $$t_{n+1} = at_n + 3n + 2, \quad t \in \mathbb{N}, \quad t_1 = -2,$$ where \(a\) is a non zero constant.

1. Given that \(\sum_{r=1}^{3} (r^3 + t_r) = 12\), determine the possible values of \(a\). [4]
2. Evaluate \(\sum_{r=8}^{31} (t_{r+1} - at_r)\). [4]

A sequence \(\{u_n\}\) is given by $$u_1 = k$$ $$u_{2n} = u_{2n-1} \times p \qquad n \geq 1$$ $$u_{2n+1} = u_{2n} \times q \qquad n \geq 1$$ where \(k\), \(p\) and \(q\) are positive constants with \(pq \neq 1\)

1. Write down the first 6 terms of this sequence. [3]
2. Show that \(\sum_{r=1}^{2n} u_r = \frac{k(1+p)(1-(pq)^n)}{1-pq}\) [6]

In part (c) \([x]\) means the integer part of \(x\), so for example \([2.73] = 2\), \([4] = 4\) and \([0] = 0\)

1. Find \(\sum_{r=1}^{\infty} 6 \times \left(\frac{4}{3}\right)^{\left[\frac{r}{2}\right]} \times \left(\frac{3}{5}\right)^{\left[\frac{r-1}{2}\right]}\) [4]

[Total 13 marks]