4.06b Method of differences: telescoping series

1

1. Express \(\frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\) in partial fractions.
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\), expressing the result as a single fraction.

7.

1. 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.
2. Hence, calculate \(\sum _ { r = 1 } ^ { 20 } ( r + 2 ) ^ { 2 }\).
   

9.

1. Given that \(A _ { r } = \frac { 1 } { r + 1 } - \frac { 2 } { r + 2 } + \frac { 1 } { r + 3 }\), show that \(A _ { r }\) can be expressed as \(\frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) }\).
2. Hence, show that \(\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { 1 } { 6 } - \frac { 1 } { ( n + 2 ) ( n + 3 ) }\).
3. Find the ratio of \(\sum _ { r = 1 } ^ { 5 } A _ { r } : \sum _ { r = 1 } ^ { 10 } A _ { r }\), giving your answer in its simplest form.

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.

4. A sequence \(\left\{ u _ { n } \right\}\), where \(n \geqslant 1\), satisfies the recurrence relation $$2 u _ { n } = u _ { n - 1 } - k n ^ { 2 } \text { where } 4 u _ { 2 } - u _ { 0 } = 27 k ^ { 2 }$$ and \(k\) is a non-zero constant.
Show that, as \(n\) becomes large, \(u _ { n }\) can be approximated by a quadratic function of the form \(a n ^ { 2 } + b n + c\) where \(a , b\) and \(c\) are constants to be determined. Please check the examination details below before entering your candidate information
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□ \section*{Thursday 14 May 2020} You may not need to use all of these tables. 3. \begin{table}[h] \end{table} 4. .

1. Prove that, for \(n \in \mathbb { Z } , n \geqslant 0\)

$$\sum _ { r = 0 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) ( r + 3 ) } = \frac { ( n + a ) ( n + b ) } { c ( n + 2 ) ( n + 3 ) }$$ where \(a\), \(b\) and \(c\) are integers to be found.

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

1. Explain why, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { 2 n } ( - 1 ) ^ { r } \mathrm { f } ( r ) = \sum _ { r = 1 } ^ { n } ( \mathrm { f } ( 2 r ) - \mathrm { f } ( 2 r - 1 ) )$$ for any function \(\mathrm { f } ( r )\).
2. Use the standard summation formulae to show that, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { 2 n } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 } = n ( 2 n + 1 ) \left( 8 n ^ { 2 } + 4 n + 5 \right)$$
3. Hence evaluate $$\sum _ { r = 14 } ^ { 50 } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 }$$

1. Prove that

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) } = \frac { n ( a n + b ) } { 12 ( n + 2 ) ( n + 3 ) }$$ where \(a\) and \(b\) are constants to be found.

1. Use the method of differences to show that

$$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 4 ) ( r + 6 ) } = \frac { n ( a n + b ) } { 30 ( n + 5 ) ( n + 6 ) }$$ where \(a\) and \(b\) are integers to be determined.

1 In this question you must show detailed reasoning. Find \(\sum _ { r = 2 } ^ { 50 } \left( \frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \right)\), expressing the answer as an exact fraction.

3 In this question you must show detailed reasoning.

1. The sequence \(\left\{ A _ { n } \right\}\) is given by \(A _ { 1 } = \sqrt { 3 }\) and \(A _ { n + 1 } = ( \sqrt { 3 } + 1 ) A _ { n }\) for \(n \geqslant 1\). Find an expression for \(A _ { n }\) in terms of \(n\).
2. The sequence \(\left\{ B _ { n } \right\}\) is given by the formula $$B _ { n } = \frac { 1 } { \sqrt { 3 } } \left( ( \sqrt { 3 } + 1 ) ^ { n } - ( \sqrt { 3 } - 1 ) ^ { n } \right) \text { for } n \geqslant 1 .$$ Explain why \(B _ { n } \rightarrow \frac { 1 } { \sqrt { 3 } } ( \sqrt { 3 } + 1 ) ^ { n }\) as \(n \rightarrow \infty\).
3. The sequence \(\left\{ C _ { n } \right\}\) converges and is defined by \(C _ { n } = \frac { A _ { n } } { B _ { n } }\) for \(n \geqslant 1\). Identify the limit of \(C n\) as \(n \rightarrow \infty\).

6

1. Find as a single algebraic fraction an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).
2. Determine the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).

7

1. Determine an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) giving your answer in the form \(\frac { 1 } { 4 } - \frac { 1 } { 2 } \mathrm { f } ( n )\).
2. Find the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

3

1. Find \(\sum _ { r = 1 } ^ { n } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\).
2. What does the sum in part (i) tend to as \(n \rightarrow \infty\) ? Justify your answer.

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 .

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

1

1. Show that $$\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } = \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$$
2. Hence find the sum of the first \(n\) terms of the series $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots$$

7

1. Use the identity \(\tan ( A - B ) = \frac { \tan A - \tan B } { 1 + \tan A \tan B }\) with \(A = ( r + 1 ) x\) and \(B = r x\) to show that $$\tan r x \tan ( r + 1 ) x = \frac { \tan ( r + 1 ) x } { \tan x } - \frac { \tan r x } { \tan x } - 1$$ (4 marks)
2. Use the method of differences to show that $$\tan \frac { \pi } { 50 } \tan \frac { 2 \pi } { 50 } + \tan \frac { 2 \pi } { 50 } \tan \frac { 3 \pi } { 50 } + \ldots + \tan \frac { 19 \pi } { 50 } \tan \frac { 20 \pi } { 50 } = \frac { \tan \frac { 2 \pi } { 5 } } { \tan \frac { \pi } { 50 } } - 20$$

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 $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$

3

1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { 4 } r ^ { 2 } ( r + 1 ) ^ { 2 }\), show that $$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = r ^ { 3 }$$
2. Use the method of differences to show that $$\sum _ { r = n } ^ { 2 n } r ^ { 3 } = \frac { 3 } { 4 } n ^ { 2 } ( n + 1 ) ( 5 n + 1 )$$