Sum from n+1 to 2n or similar range

3

1. Use the method of differences to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.
2. Deduce the value of \(\sum _ { \mathrm { r } = 1 } ^ { \infty } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }\).
3. Find also \(\sum _ { \mathrm { r } = \mathrm { n } } ^ { \mathrm { n } ^ { 2 } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }\) in terms of \(n\) and \(k\).

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 \(n\).

4

1. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-08_2720_35_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-09_2723_35_101_20} It is given that \(\sum _ { r = 1 } ^ { \infty } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) } = \frac { 1 } { 3 }\).
2. Find the value of \(k\).
3. Hence find \(\sum _ { r = n } ^ { n ^ { 2 } } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\). $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x y$$ has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\).
   The curve \(C\) has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
4. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-10_2716_35_108_2012}
5. Find the area of the region enclosed by \(C\).
6. Find the maximum distance of a point on \(C\) from the initial line.

4

1. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-08_2715_35_110_2012} \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-09_2723_35_101_20} It is given that \(\sum _ { r = 1 } ^ { \infty } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) } = \frac { 1 } { 3 }\).
2. Find the value of \(k\).
3. Hence find \(\sum _ { r = n } ^ { n ^ { 2 } } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\). $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x y$$ has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\).
   The curve \(C\) has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
4. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-10_2716_35_108_2012}
5. Find the area of the region enclosed by \(C\).
6. Find the maximum distance of a point on \(C\) from the initial line.

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 $$\sum _ { r = 10 } ^ { 20 } r ( 2 r + 1 ) ( 3 r + 1 )$$

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. (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.

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 $$\sum _ { r = 25 } ^ { 49 } \left( r ^ { 2 } ( r + 1 ) + 2 \right)$$

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}

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 \(n\)

$$\sum _ { r = 0 } ^ { n } ( r + 1 ) ( r + 2 ) = \frac { 1 } { 3 } ( n + 1 ) ( n + 2 ) ( n + 3 )$$ (b) Hence determine the value of $$10 \times 11 + 11 \times 12 + 12 \times 13 + \ldots + 100 \times 101$$

5. (a) 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 for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r - 1 ) ( r - 3 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n - 1 ) ( 3 n - 10 )$$ (b) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n + 1 } r ( r - 1 ) ( r - 3 ) = \frac { 1 } { 12 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.

6. (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 \(\sum _ { r = 21 } ^ { 40 } r ( r + 1 ) ( r + 3 )\)

5. (a) Use the results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), to prove that $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 5 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 2 ) ( n + 7 )$$ for all positive integers \(n\).
(b) Hence, or otherwise, find the value of $$\sum _ { r = 20 } ^ { 50 } r ( r + 1 ) ( r + 5 )$$

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 \(\sum _ { r = 21 } ^ { 40 } r ( r + 1 ) ( r + 3 )\).

7. (a) Use the results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n ( 2 n + 1 ) ( 2 n - 1 )$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 3 n } ( 2 r - 1 ) ^ { 2 } = \frac { 2 } { 3 } n \left( a n ^ { 2 } + b \right)$$ where \(a\) and \(b\) are integers to be found.

7. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { n ( n + 1 ) ( 3 n + 2 ) ( n - 1 ) } { 12 }$$ for all positive integers \(n\).
(b) Hence find the sum of the series $$10 ^ { 2 } \times 9 + 11 ^ { 2 } \times 10 + 12 ^ { 2 } \times 11 + \ldots + 50 ^ { 2 } \times 49$$

5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r + 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 9 n + 26 \right)$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 3 n } ( r + 2 ) ( r + 3 ) = \frac { 2 } { 3 } n \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be found.

1. (a) Using the formula for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) write down, in terms of \(n\) only, an expression for

$$\sum _ { r = 1 } ^ { 3 n } r ^ { 2 }$$ (b) Show that, for all integers \(n\), where \(n > 0\) $$\sum _ { r = 2 n + 1 } ^ { 3 n } r ^ { 2 } = \frac { n } { 6 } \left( a n ^ { 2 } + b n + c \right)$$ where the values of the constants \(a\), \(b\) and \(c\) are to be found.

6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 6 \right) = \frac { 1 } { 2 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$ (b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 50 } r \left( 2 r ^ { 2 } - 6 \right)\).

5. (a) Show that \(\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 1 \right) = \frac { 1 } { 3 } ( n - 2 ) n ( n + 2 )\).
(b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 40 } \left( r ^ { 2 } - r - 1 \right)\).

1. (a) Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions.
   (b) Hence prove, by the method of differences, that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 2 ) } = \frac { n ( a n + b ) } { 4 ( n + 1 ) ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.
(c) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } \frac { 1 } { r ( r + 2 ) } = \frac { n ( 4 n + 5 ) } { 4 ( n + 1 ) ( n + 2 ) ( 2 n + 1 ) }$$

1. (a) Show that, for \(r > 0\)

$$\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } \equiv \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$$ (b) Hence prove that, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ( n + 2 ) } { ( n + 1 ) ^ { 2 } }$$ (c) Show that, for \(n \in \mathbb { N } , n > 1\) $$\sum _ { r = n } ^ { 3 n } \frac { 6 r + 3 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { a n ^ { 2 } + b n + c } { n ^ { 2 } ( 3 n + 1 ) ^ { 2 } }$$ where \(a , b\) and \(c\) are constants to be found.

5 Use the method of differences to show that \(\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 1 } { 6 } - \frac { 1 } { 2 ( 2 N + 3 ) }\). Deduce that \(\sum _ { r = N + 1 } ^ { 2 N } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) } < \frac { 1 } { 8 N }\).

4

1. Use the method of differences to show that \(\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r - 2 ) } = \frac { 1 } { 3 } - \frac { 1 } { 3 ( 3 N + 1 ) }\).
2. Find the limit, as \(N \rightarrow \infty\), of \(\sum _ { r = N + 1 } ^ { N ^ { 2 } } \frac { N } { ( 3 r + 1 ) ( 3 r - 2 ) }\).

2 Given that $$u _ { n } = \frac { 1 } { n ^ { 2 } - n + 1 } - \frac { 1 } { n ^ { 2 } + n + 1 } ,$$ find \(S _ { N } = \sum _ { n = N + 1 } ^ { 2 N } u _ { n }\) in terms of \(N\). Find a number \(M\) such that \(S _ { N } < 10 ^ { - 20 }\) for all \(N > M\).