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

3 Let \(S _ { n } = 2 ^ { 2 } + 6 ^ { 2 } + 10 ^ { 2 } + \ldots + ( 4 n - 2 ) ^ { 2 }\).

1. Use standard results from the List of Formulae (MF19) to show that \(S _ { n } = \frac { 4 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)\).
2. Express \(\frac { \mathrm { n } } { \mathrm { S } _ { \mathrm { n } } }\) in partial fractions and find \(\sum _ { \mathrm { n } = 1 } ^ { \mathrm { N } } \frac { \mathrm { n } } { \mathrm { S } _ { \mathrm { n } } }\) in terms of \(N\).
3. Deduce the value of \(\sum _ { n = 1 } ^ { \infty } \frac { n } { S _ { n } }\).

2

1. Use standard results from the List of formulae (MF19) to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( 1 - \mathrm { r } - \mathrm { r } ^ { 2 } \right)\) in terms of \(n\),
   simplifying your answer. simplifying your answer.
2. Show that $$\frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) } = \frac { r + 1 } { ( r + 1 ) ^ { 2 } + 1 } - \frac { r } { r ^ { 2 } + 1 }$$ and hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) }\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) }\).

3

1. Prove by mathematical induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 5 r ^ { 4 } + r ^ { 2 } \right) = \frac { 1 } { 2 } n ^ { 2 } ( n + 1 ) ^ { 2 } ( 2 n + 1 )$$
2. Use the result given in part (a) together with the List of formulae (MF19) to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { r } ^ { 4 }\) in terms of \(n\), fully factorising your answer.

4 Let \(\mathrm { u } _ { \mathrm { r } } = \mathrm { e } ^ { \mathrm { rx } } \left( \mathrm { e } ^ { 2 \mathrm { x } } - 2 \mathrm { e } ^ { \mathrm { x } } + 1 \right)\).

1. Using the method of differences, or otherwise, find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { u } _ { \mathrm { r } }\) in terms of \(n\) and \(x\).
2. Deduce the set of non-zero values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists.
3. Using a standard result from the list of formulae (MF19), find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \ln \mathrm { u } _ { \mathrm { r } }\) in terms of \(n\) and \(x\).

2 The cubic equation \(x ^ { 3 } + 4 x ^ { 2 } + 6 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
2. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 2 } + ( \beta + r ) ^ { 2 } + ( \gamma + r ) ^ { 2 } \right) = n \left( n ^ { 2 } + a n + b \right)$$ where \(a\) and \(b\) are constants to be determined. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

2

1. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 6 r - 5 \right) = a n ^ { 3 } + b n ^ { 2 } + c n$$ where \(a\), \(b\) and \(c\) are integers to be determined.
2. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 6 r ^ { 2 } + 6 r - 5 } { r ^ { 2 } + r }\) in terms of \(n\).
3. Find also \(\sum _ { r = n + 1 } ^ { 2 n } \frac { 6 r ^ { 2 } + 6 r - 5 } { r ^ { 2 } + r }\) in terms of \(n\).

3

1. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { N } r ( r + 1 ) ( 3 r + 4 ) = \frac { 1 } { 12 } N ( N + 1 ) ( N + 2 ) ( 9 N + 19 )$$
2. Express \(\frac { 3 r + 4 } { r ( r + 1 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { N } \frac { 3 r + 4 } { r ( r + 1 ) } \left( \frac { 1 } { 4 } \right) ^ { r + 1 }$$ in terms of \(N\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) } \left( \frac { 1 } { 4 } \right) ^ { r + 1 }\).

2

1. Use standard results from the List of Formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } ( 7 r + 1 ) ( 7 r + 8 ) = a n ^ { 3 } + b n ^ { 2 } + c n$$ where \(a , b\) and \(c\) are constants to be determined.
2. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 7 r + 1 ) ( 7 r + 8 ) }\) in terms of \(n\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 7 r + 1 ) ( 7 r + 8 ) }\).

2

1. Use standard results from the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 2 )\) in terms of \(n\),
   fully factorising your answer. fully factorising your answer.
2. Express \(\frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

4 The cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 3 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 1\).
3. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 3 } + ( \beta + r ) ^ { 3 } + ( \gamma + r ) ^ { 3 } \right) = n + \frac { 1 } { 4 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are constants to be determined.

3

1. By considering \(( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ Let \(S _ { n } = 1 ^ { 2 } + 3 \times 2 ^ { 2 } + 3 ^ { 2 } + 3 \times 4 ^ { 2 } + 5 ^ { 2 } + 3 \times 6 ^ { 2 } + \ldots + \left( 2 + ( - 1 ) ^ { n } \right) n ^ { 2 }\).
2. Show that \(\mathrm { S } _ { 2 \mathrm { n } } = \frac { 1 } { 3 } \mathrm { n } ( 2 \mathrm { n } + 1 ) ( \mathrm { an } + \mathrm { b } )\), where \(a\) and \(b\) are integers to be determined.
3. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 3 } }\).

1

1. Use the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } r ( r + 2 )\) in terms of \(n\), simplifying your answer.
2. Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions and hence find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { \mathrm { r } ( \mathrm { r } + 2 ) }\) in terms of \(n\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 2 ) }\).

1

1. By considering \(( r + 1 ) ^ { 2 } - r ^ { 2 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
2. Given that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { r } + \mathrm { a } ) = \mathrm { n }\), find \(a\) in terms of \(n\).

1

1. Use standard results from the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } + 3 r + 1 \right)\) in terms of \(n\), simplifying your answer.
2. Show that $$\frac { 1 } { r ^ { 3 } } - \frac { 1 } { ( r + 1 ) ^ { 3 } } = \frac { 3 r ^ { 2 } + 3 r + 1 } { r ^ { 3 } ( r + 1 ) ^ { 3 } }$$ and hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 3 r ^ { 2 } + 3 r + 1 } { r ^ { 3 } ( r + 1 ) ^ { 3 } }\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r ^ { 2 } + 3 r + 1 } { r ^ { 3 } ( r + 1 ) ^ { 3 } }\).

4 \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_408_1433_296_315} The diagram shows the curve with equation \(y = x ^ { - 2 }\) for \(2 \leqslant x \leqslant N\) together with a set of ( \(N - 2\) ) rectangles of unit width.

1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } } > \frac { 3 } { 2 } - \frac { 1 } { N } + \frac { 1 } { N ^ { 2 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_2718_35_141_2012}
2. Use a similar method to find, in terms of \(N\), an upper bound for \(\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } }\).
3. Deduce lower and upper bounds for \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\).

6

1. Show that \(\sum _ { r = 1 } ^ { n } z ^ { 4 r } = \frac { z ^ { 4 n + 2 } - z ^ { 2 } } { z ^ { 2 } - z ^ { - 2 } }\), for \(z ^ { 2 } \neq z ^ { - 2 }\).
2. By letting \(z = \cos \theta + \mathrm { i } \sin \theta\), show that, if \(\sin 2 \theta \neq 0\), $$\sum _ { r = 1 } ^ { n } \sin ( 4 r \theta ) = \frac { \cos 2 \theta - \cos ( 4 n + 2 ) \theta } { 2 \sin 2 \theta }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-12_2718_35_143_2012}

5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } \left( 9 r ^ { 2 } - 4 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 6 n - 1 )$$ for all positive integers \(n\). Given that $$\sum _ { r = 1 } ^ { 12 } \left( 9 r ^ { 2 } - 4 r + k \left( 2 ^ { r } \right) \right) = 6630$$ (b) find the exact value of the constant \(k\).

7. Given that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } ( r + a ) ( r + b ) = \frac { 1 } { 6 } n ( 2 n + 11 ) ( n - 1 )$$ where \(a\) and \(b\) are constants and \(a > b\),

1. find the value of \(a\) and the value of \(b\).
2. Find the value of $$\sum _ { r = 9 } ^ { 20 } ( r + a ) ( r + b )$$

5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 3 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 3 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be found. Given that $$\sum _ { r = 5 } ^ { 10 } \left( 8 r ^ { 3 } - 3 r + k r ^ { 2 } \right) = 22768$$ (b) find the exact value of the constant \(k\).

4. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( 2 r + 1 ) ( 3 r + 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
(b) Hence find the value of $$\sum _ { r = 10 } ^ { 20 } r ( 2 r + 1 ) ( 3 r + 1 )$$

1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) 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) Given that $$\sum _ { r = 5 } ^ { 25 } r ^ { 2 } ( r + 1 ) + \sum _ { r = 1 } ^ { k } 3 ^ { r } = 140543$$ find the value of the integer \(k\).

5. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that $$\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 ) = \frac { n } { 6 } ( n + 7 ) ( 2 n + 7 )$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) ( r + 5 ) = \frac { 7 n } { 6 } ( n + 1 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be determined.

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.

9. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 2 ) = \frac { n ( n + 1 ) ( n + 2 ) ( n + 3 ) } { 4 }$$ (ii) Prove by induction that, $$4 ^ { n } + 6 n + 8 \text { is divisible by } 18$$ for all positive integers \(n\). \includegraphics[max width=\textwidth, alt={}, center]{df5ab400-5cb1-4b51-8b0a-52dc3587f81a-16_62_44_2476_1889}