1.04g Sigma notation: for sums of series

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that for all positive integers \(n\) $$\sum _ { r = 1 } ^ { n } ( 7 r - 5 ) ^ { 2 } = \frac { n } { 6 } ( 7 n + 1 ) ( A n + B )$$ where \(A\) and \(B\) are integers to be determined.

1. (a) Use the standard results for summations to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n - 2 )$$ (b) Hence show that, for all positive integers \(n\), $$\sum _ { r = n } ^ { 2 n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n - 1 ) ( a n + b ) ( c n + d )$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be determined.

1. Find the value of

$$\sum _ { r = 1 } ^ { 200 } ( r + 1 ) ( r - 1 )$$

6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { n } { a } ( n + 1 ) ( n + 2 ) ( 3 n + b )$$ where \(a\) and \(b\) are integers to be found.
(b) Hence find the value of $$\sum _ { r = 25 } ^ { 49 } \left( r ^ { 2 } ( r + 1 ) + 2 \right)$$

1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = \frac { n } { a } ( n + 1 ) ( n + b )$$ where \(a\) and \(b\) are integers to be found.

4. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$ for all positive integers \(n\).
(b) Hence find the exact value of the sum of the squares of the odd numbers between 200 and 500 \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-13_2255_50_314_34}

2. (a) Show, using the formulae for \(\sum r\) and \(\sum r ^ { 2 }\), that $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 4 r - 1 \right) = n ( n + 2 ) ( 2 n + 1 )$$ (b) Hence, or otherwise, find the value of \(\sum _ { r = 11 } ^ { 20 } \left( 6 r ^ { 2 } + 4 r - 1 \right)\).

1. Show, using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), that

$$\sum _ { r = 1 } ^ { n } 3 ( 2 r - 1 ) ^ { 2 } = n ( 2 n + 1 ) ( 2 n - 1 ) , \text { for all positive integers } n .$$

6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n - 1 ) ( n + a )$$ where \(a\) is an integer to be determined.
(b) Hence find the value of \(n\), where \(n > 1\), that satisfies $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 1 ) = 10 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$

2. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), show that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 3 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + k ) ,$$ where \(k\) is a constant to be found.
(b) Hence evaluate \(\sum _ { r = 21 } ^ { 40 } r ( r + 1 ) ( r + 3 )\).

A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{array} { l l } a _ { n + 1 } = 4 a _ { n } - 3 , & n \geqslant 1 \\ a _ { 1 } = k , & \text { where } k \text { is a positive integer. } \end{array}$$

1. Write down an expression for \(a _ { 2 }\) in terms of \(k\). Given that \(\sum _ { r = 1 } ^ { 3 } a _ { r } = 66\)
2. find the value of \(k\).

An arithmetic series has first term \(a\) and common difference \(d\).

1. Prove that the sum of the first \(n\) terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$$ A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week \(N\).
2. Find the value of \(N\) The company then plans to continue to make 600 mobile phones each week.
3. Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.
   
   \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-16_2673_1948_107_118}

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

7

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 \(y = 2 ^ { x }\).

4

1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).

3

1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.

5 A sequence is defined by \(a _ { k } = 5 k + 1\), for \(k = 1,2,3 \ldots\)

1. Write down the first three terms of the sequence.
2. Evaluate \(\sum _ { k = 1 } ^ { 100 } a _ { k }\).

1. Evaluate

$$\sum _ { r = 1 } ^ { 12 } \left( 5 \times 2 ^ { r } \right)$$

1. (i) Evaluate

$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (ii) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where \(k\) is a rational constant to be found.

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

2 The \(n\)th term of a sequence, \(u _ { n }\), is given by $$u _ { n } = 12 - \frac { 1 } { 2 } n .$$

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\). State what type of sequence this is.
2. Find \(\sum _ { n = 1 } ^ { 30 } u _ { n }\).

7

1. Evaluate \(\sum _ { r = 2 } ^ { 5 } \frac { 1 } { r - 1 }\).
2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum _ { r = 2 } ^ { a } \mathrm { f } ( r )\) where \(\mathrm { f } ( r )\) and \(a\) are to be determined.

8

1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.

9

1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).