4.06a Summation formulae: sum of r, r^2, r^3

7. (a) Find an expression for \(\sum _ { r = 1 } ^ { 2 m } ( r + 2 ) ^ { 2 }\) in the form \(\frac { 1 } { 3 } m \left( a m ^ { 2 } + b m + c \right)\), where \(a , b , c\) are integers whose values are to be determined.
(b) Hence, calculate \(\sum _ { r = 1 } ^ { 20 } ( r + 2 ) ^ { 2 }\).

4. The positive integer \(N\) is such that \(1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + N ^ { 2 } = ( 3 N - 2 ) ^ { 2 }\). Write down and simplify an equation satisfied by \(N\). Hence find the possible values of \(N\).

10. Gareth is investigating a series involving cube numbers. His series is $$1 ^ { 3 } - 2 ^ { 3 } + 3 ^ { 3 } - 4 ^ { 3 } + 5 ^ { 3 } - 6 ^ { 3 } + 7 ^ { 3 } - \ldots$$ Gareth continues his series and ends with an odd number.
Find and simplify an expression for the sum of Gareth's series in terms of \(k\), where \(k\) is the number of odd numbers in his series.

5. Given that $$\sum _ { r = k } ^ { 76 } ( r - 31 ) = 980$$ show that there are two possible values of \(k\).
\section*{PLEASE DO NOT WRITE ON THIS PAGE}

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

$$\sum _ { r = 1 } ^ { n } ( 3 r - 2 ) ^ { 2 } = \frac { 1 } { 2 } n \left[ 6 n ^ { 2 } - 3 n - 1 \right]$$ for all positive integers \(n\).
(b) Hence find any values of \(n\) for which $$\sum _ { r = 5 } ^ { n } ( 3 r - 2 ) ^ { 2 } + 103 \sum _ { r = 1 } ^ { 28 } r \cos \left( \frac { r \pi } { 2 } \right) = 3 n ^ { 3 }$$

5. \begin{figure}[h] \end{figure} A block has length \(( r + 2 ) \mathrm { cm }\), width \(( r + 1 ) \mathrm { cm }\) and height \(r \mathrm {~cm}\), as shown in Figure 2.
In a set of \(n\) such blocks, the first block has a height of 1 cm , the second block has a height of 2 cm , the third block has a height of 3 cm and so on.

1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that the total volume, \(V\), of all \(n\) blocks in the set is given by $$V = \frac { n } { 4 } ( n + 1 ) ( n + 2 ) ( n + 3 ) \quad n \geqslant 1$$ Given that the total volume of all \(n\) blocks is $$\left( n ^ { 4 } + 6 n ^ { 3 } - 11710 \right) \mathrm { cm } ^ { 3 }$$
2. determine how many blocks make up the set.

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

$$\sum _ { r = 1 } ^ { n } ( 5 r - 2 ) ^ { 2 } = \frac { 1 } { 6 } n \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
(b) Hence determine the value of \(k\) for which $$\sum _ { r = 1 } ^ { k } ( 5 r - 2 ) ^ { 2 } = 94 k ^ { 2 }$$

1. (a) Prove by induction that, for all positive integers \(n\),

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

1. (a) Use the standard summation formulae to show that, for \(n \in \mathbb { N }\),

$$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 17 r - 25 \right) = n \left( n ^ { 2 } - A n - B \right)$$ where \(A\) and \(B\) are integers to be determined.
(b) Explain why, for \(k \in \mathbb { N }\), $$\sum _ { r = 1 } ^ { 3 k } r \tan ( 60 r ) ^ { \circ } = - k \sqrt { 3 }$$ Using the results from part (a) and part (b) and showing all your working,
(c) determine any value of \(n\) that satisfies $$\sum _ { r = 5 } ^ { n } \left( 3 r ^ { 2 } - 17 r - 25 \right) = 15 \left[ \sum _ { r = 6 } ^ { 3 n } r \tan ( 60 r ) ^ { \circ } \right] ^ { 2 }$$

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

$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { n } { 3 } \left( a n ^ { 2 } - 1 \right)$$ where \(a\) is a constant to be determined.
(b) Hence determine the sum of the squares of all positive odd three-digit integers.

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

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be determined.
(b) Hence show that, for all positive integers \(k\), $$\sum _ { r = k + 1 } ^ { 3 k } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 3 } k ( 3 k + 1 ) \left( A k ^ { 2 } + B k + C \right)$$ where \(A , B\) and \(C\) are integers to be determined.
(c) Hence, using algebra and making your method clear, determine the value of \(k\) for which $$25 \sum _ { r = k + 1 } ^ { 3 k } r ^ { 2 } ( r + 1 ) = 192 k ^ { 3 } ( 3 k + 1 )$$

1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)

$$\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r + 2 ) = n ( n + 2 ) ( n + 3 )$$ (ii) Prove by induction that for all positive odd integers \(n\) $$f ( n ) = 4 ^ { n } + 5 ^ { n } + 6 ^ { n }$$ is divisible by 15

1. Prove by induction that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$

1. In this question you may assume the results for

$$\sum _ { r = 1 } ^ { n } r ^ { 3 } , \sum _ { r = 1 } ^ { n } r ^ { 2 } \text { and } \sum _ { r = 1 } ^ { n } r$$

1. Show that the sum of the cubes of the first \(n\) positive odd numbers is $$n ^ { 2 } \left( 2 n ^ { 2 } - 1 \right)$$ The sum of the cubes of 10 consecutive positive odd numbers is 99800
2. Use the answer to part (a) to determine the smallest of these 10 consecutive positive odd numbers.

1. The number of visits to a website, in any particular month, is modelled as the number of visits received in the previous month plus \(k\) times the number of visits received in the month before that, where \(k\) is a positive constant.

Given that \(V _ { n }\) is the number of visits to the website in month \(n\),

1. write down a general recurrence relation for \(V _ { n + 2 }\) in terms of \(V _ { n + 1 } , V _ { n }\) and \(k\). For a particular website you are given that
   • \(k = 0.24\)
   • In month 1 , there were 65 visits to the website.
   • In month 2 , there were 71 visits to the website.
   • Show that
   $$V _ { n } = 50 ( 1.2 ) ^ { n } - 25 ( - 0.2 ) ^ { n }$$ This model predicts that the number of visits to this website will exceed one million for the first time in month \(N\).
2. Find the value of \(N\).

1. Solve the recurrence system

$$\begin{gathered} u _ { 1 } = 1 \quad u _ { 2 } = 4 \\ 9 u _ { n + 2 } - 12 u _ { n + 1 } + 4 u _ { n } = 3 n \end{gathered}$$

1. (a) Prove by induction that for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ (b) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r + 6 ) ( r - 6 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n - 8 ) ( n + 9 )$$ (c) Hence find the value of \(n\) that satisfies $$\sum _ { r = 1 } ^ { n } r ( r + 6 ) ( r - 6 ) = 17 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$

7 In this question you must show detailed reasoning. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) is defined by \(u _ { n } = 25 \times 0.6 ^ { n }\).
Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { \infty } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } < 10 ^ { - 4 }\).

4

1. Find $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 6 r \right)$$ expressing your answer in the form $$k n ( n + 1 ) ( n + p ) ( n + q )$$ where \(k\) is a fraction and \(p\) and \(q\) are integers.
2. It is given that $$S = \sum _ { r = 1 } ^ { 1000 } \left( r ^ { 3 } - 6 r \right)$$ Without calculating the value of \(S\), show that \(S\) is a multiple of 2008 .

8

1. Show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r$$ can be expressed in the form $$k n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(k\) is a rational number and \(a , b\) and \(c\) are integers.
2. Show that there is exactly one positive integer \(n\) for which $$\sum _ { r = 1 } ^ { n } r ^ { 3 } + \sum _ { r = 1 } ^ { n } r = 8 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$

3

1. Use the formulae $$\begin{gathered} \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 ) \\ \sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 } \end{gathered}$$ and $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { 1 } { 12 } n \left( n ^ { 2 } - 1 \right) ( 3 n + 2 )$$ (4 marks)
2. Use the result from part (a) to find the value of $$\sum _ { r = 4 } ^ { 11 } r ^ { 2 } ( r - 1 )$$ (3 marks)

3 Show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r \right) = k n ( n + 1 ) ( n - 1 )$$ where \(k\) is a rational number.

4

1. Prove by induction that $$2 + ( 3 \times 2 ) + \left( 4 \times 2 ^ { 2 } \right) + \ldots + ( n + 1 ) 2 ^ { n - 1 } = n 2 ^ { n }$$ for all integers \(n \geqslant 1\).
2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n } \left( 2 ^ { n + 1 } - 1 \right)$$

2

1. Show that $$( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 } = 24 r ^ { 2 } + 2$$
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$