1.04g Sigma notation: for sums of series

Showing all your working, evaluate

1. \(\sum_{r=3}^{50} (4r + 5)\) [4]
2. \(\sum_{r=2}^{\infty} \left(540 \times \left(\frac{1}{3}\right)^r\right)\). [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]

A series is given by $$\sum_{r=1}^k 9^{r-1}$$

1. Write down a formula for the sum of this series. [1]
2. Prove by induction that \(P(n) = 9^n - 8n - 1\) is divisible by 64 if \(n\) is a positive integer greater than 1. [4]

A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find

1. the number of houses built in 1986, the first year of the building programme, [5]
2. the total number of houses built in the 25 years of the programme. [2]

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]

Given that \(p\) is a positive constant,

1. show that $$\sum_{n=1}^{11} \ln(p^n) = k \ln p$$ where \(k\) is a constant to be found, [2]
2. show that $$\sum_{n=1}^{11} \ln(8p^n) = 33\ln(2p^2)$$ [2]
3. Hence find the set of values of \(p\) for which $$\sum_{n=1}^{11} \ln(8p^n) < 0$$ giving your answer in set notation. [2]

Find \(\sum_{r=1}^{n}(2r^2 - 1)\), expressing your answer in fully factorised form. [4]

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]