1.04g Sigma notation: for sums of series

11 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).

12 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\). 12 A sequence begins $$\begin{array} { l l l l l l l l l l l } 1 & 3 & 5 & 3 & 1 & 3 & 5 & 3 & 1 & 3 & \ldots \end{array}$$ and continues in this pattern.

1. Find the 55th term of this sequence, showing your method.
2. Find the sum of the first 55 terms of the sequence.

1 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).

5 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).

6

1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
3. Sketch the curve with equation \(y = 2 ^ { x }\).

7

1. By using a set of rectangles of unit width to approximate an area under the curve \(y = \frac { 1 } { x }\), show that \(\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x }\) is infinite.
2. By using a set of rectangles of unit width to approximate an area under the curve \(y = \frac { 1 } { x ^ { 2 } }\), find an upper limit for the series \(\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x ^ { 2 } }\).

1.(a)By considering the series $$1 + t + t ^ { 2 } + t ^ { 3 } + \ldots + t ^ { n }$$ or otherwise,sum the series $$1 + 2 t + 3 t ^ { 2 } + 4 t ^ { 3 } + \ldots + n t ^ { n - 1 }$$ for \(t \neq 1\) .
(b)Hence find and simplify an expression for $$1 + 2 \times 3 + 3 \times 3 ^ { 2 } + 4 \times 3 ^ { 3 } + \ldots + 2001 \times 3 ^ { 2000 }$$ (c)Write down an expression for both the sums of the series in part(a)for the case where \(t = 1\) .

5.(a)Show that $$\sum _ { r = 0 } ^ { n } x ^ { - r } = \frac { x } { x - 1 } - \frac { x ^ { - n } } { x - 1 } \quad \text { where } x \neq 0 \text { and } x \neq 1$$ (b)Hence find an expression in terms of \(x\) and \(n\) for \(\sum _ { r = 0 } ^ { n } r x ^ { - ( r + 1 ) }\) for \(x \neq 0\) and \(x \neq 1\) Simplify your answer.
(c)Find \(\sum _ { r = 0 } ^ { n } \left( \frac { 3 + 5 r } { 2 ^ { r } } \right)\) Give your answer in the form \(a - \frac { b + c n } { 2 ^ { n } }\) ,where \(a , b\) and \(c\) are integers.

5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$a _ { n + 1 } = 5 a _ { n } - 3 , \quad n \geqslant 1$$ Given that \(a _ { 2 } = 7\),

1. find the value of \(a _ { 1 }\)
2. Find the value of \(\sum _ { r = 1 } ^ { 4 } a _ { r }\)

3 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 24 - \frac { 2 } { 3 } n$$

1. Write down the exact values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find the value of \(k\) such that \(u _ { k } = 0\).
3. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).

8 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 \(u _ { 5 } = 20\).
2. The \(n\)th term of the sequence can be written in the form \(u _ { n } = p n + q\). State the values of \(p\) and \(q\).
3. State what type of sequence it is.
4. Find the value of \(N\) such that \(\sum _ { n = 1 } ^ { 2 N } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } = 1256\).

2 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by \(u _ { n } = 3 n + 2\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. State what type of sequence \(S\) is.
3. Find \(\sum _ { n = 101 } ^ { 200 } u _ { n }\).

2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 7 \text { and } u _ { n + 1 } = u _ { n } + 4 \text { for } n \geqslant 1 .$$

1. Show that \(u _ { 17 } = 71\).
2. Show that \(\sum _ { n = 1 } ^ { 35 } u _ { n } = \sum _ { n = 36 } ^ { 50 } u _ { n }\).

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 5 n + 1\).

1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Evaluate \(\sum _ { n = 1 } ^ { 40 } u _ { n }\). Another sequence \(w _ { 1 } , w _ { 2 } , w _ { 3 } , \ldots\) is defined by \(w _ { 1 } = 2\) and \(w _ { n + 1 } = 5 w _ { n } + 1\).
3. Find the value of \(p\) such that \(u _ { p } = w _ { 3 }\).

2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 3 n - 1\), for \(n \geqslant 1\).

1. Find the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).

7 In an arithmetic progression the first term is 5 and the common difference is 3 . The \(n\)th term of the progression is denoted by \(u _ { n }\).

1. Find the value of \(u _ { 20 }\).
2. Show that \(\sum _ { n = 10 } ^ { 20 } u _ { n } = 517\).
3. Find the value of \(N\) such that \(\sum _ { n = N } ^ { 2 N } u _ { n } = 2750\).

3 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).

1 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\).

1 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).

2 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { k + 1 } = \frac { 10 } { u _ { k } ^ { 2 } }\).
Calculate \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).

3 Find \(\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 6 r ^ { 2 } + 2 r \right)\), expressing your answer in a fully factorised form.

4 Find \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 2 )\), expressing your answer in a fully factorised form.

7

1. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 4 } - r ^ { 4 } \right\} = ( n + 1 ) ^ { 4 } - 1$$
2. Show that \(( r + 1 ) ^ { 4 } - r ^ { 4 } \equiv 4 r ^ { 3 } + 6 r ^ { 2 } + 4 r + 1\).
3. Hence show that $$4 \sum _ { r = 1 } ^ { n } r ^ { 3 } = n ^ { 2 } ( n + 1 ) ^ { 2 }$$

5 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } ( - 1 ) ^ { r - 1 } r ^ { 2 }\).

1. Use the standard result for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) given in the List of Formulae (MF10) to show that $$S _ { 2 n } = - n ( 2 n + 1 )$$
2. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 2 } }\) and find \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n + 1 } } { n ^ { 2 } }\).

4. (a) Show that \(\sum _ { r = 1 } ^ { 20 } \left( 2 ^ { r - 1 } - 3 - 4 r \right) = 1047675\) (b) A sequence has \(n\)th term \(u _ { n } = \sin \left( 90 n ^ { \circ } \right) n \geq 1\)

1. Find the order of the sequence.
2. Find \(\sum _ { r = 1 } ^ { 222 } u _ { r }\)