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

4 Find \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r - 2 )\), expressing your answer in a fully factorised form.

4 Given that \(\sum _ { r = 1 } ^ { n } \left( a r ^ { 3 } + b r \right) \equiv n ( n - 1 ) ( n + 1 ) ( n + 2 )\), find the values of the constants \(a\) and \(b\).

4 Find \(\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right)\), expressing your answer in a fully factorised form.

1 Evaluate \(\sum _ { r = 101 } ^ { 250 } r ^ { 3 }\).

4 Find \(\sum _ { r = 1 } ^ { 2 n } \left( 3 r ^ { 2 } - \frac { 1 } { 2 } \right)\), expressing your answer in a fully factorised form.

4 Find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 2 \right)\), expressing your answer in a fully factorised form.

8

1. Show that \(\sum _ { r = n } ^ { 2 n } r ^ { 3 } = \frac { 3 } { 4 } n ^ { 2 } ( n + 1 ) ( 5 n + 1 )\).
2. Hence find \(\sum _ { r = n } ^ { 2 n } r \left( r ^ { 2 } - 2 \right)\), giving your answer in a fully factorised form.

2 Find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 5 \right)\), expressing your answer in a fully factorised form.

1 Find \(\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r - 1 )\), giving your answer in a fully factorised form.

6 Using the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) show that $$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$

5 Use standard series formulae to show that \(\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r - 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } - 19 \right)\).

5 Use standard series formulae to show that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 3 - 4 r ) = \frac { 1 } { 2 } n ( n + 1 ) \left( 1 - 2 n ^ { 2 } \right)\).

4 Using the standard summation formulae, find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 )\). Give your answer in a fully factorised form.

8

1. Use standard series formulae to show that $$\sum _ { r = 1 } ^ { n } [ r ( r - 1 ) - 1 ] = \frac { 1 } { 3 } n ( n + 2 ) ( n - 2 )$$
2. Prove (*) by mathematical induction.

1 Use standard series formulae to find \(\sum _ { r = 1 } ^ { n } r ( r - 2 )\), factorising your answer as far as possible.

5

1. Show that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( 2 \mathrm { r } - 1 ) = \mathrm { n } ^ { 2 }\).
2. Show that \(\frac { \sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( 2 \mathrm { r } - 1 ) } { \sum _ { \mathrm { r } = \mathrm { n } + 1 } ^ { 2 \mathrm { n } } ( 2 \mathrm { r } - 1 ) } = \mathrm { k }\), where \(k\) is a constant to be determined.

4

1. Use standard series to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - p ) = \frac { 1 } { 6 } n ( n + 1 ) \left( 3 n ^ { 2 } + ( 3 - 2 p ) n - p \right) ,$$ where \(p\) is a constant.
2. Given that the coefficients of \(n ^ { 3 }\) and \(n ^ { 4 }\) in the expression for \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - p )\) are equal, find the value of \(p\).

7 Prove by induction that $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 5 } + r ^ { 3 } \right) = \frac { 1 } { 2 } n ^ { 3 } ( n + 1 ) ^ { 3 }$$ for all \(n \geqslant 1\). Use this result together with the List of Formulae (MF10) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 5 } = \frac { 1 } { 12 } n ^ { 2 } ( n + 1 ) ^ { 2 } \mathrm { Q } ( n )$$ where \(\mathrm { Q } ( n )\) is a quadratic function of \(n\) which is to be determined.

4 The sum \(S _ { N }\) is defined by \(S _ { N } = \sum _ { n = 1 } ^ { N } n ^ { 5 }\). Using the identity $$\left( n + \frac { 1 } { 2 } \right) ^ { 6 } - \left( n - \frac { 1 } { 2 } \right) ^ { 6 } \equiv 6 n ^ { 5 } + 5 n ^ { 3 } + \frac { 3 } { 8 } n$$ find \(S _ { N }\) in terms of \(N\). [You need not simplify your result.] Hence find \(\lim _ { N \rightarrow \infty } N ^ { - \lambda } S _ { N }\), for each of the two cases

1. \(\lambda = 6\),
2. \(\lambda > 6\).

1 Find \(2 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }\). Hence find \(1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 n ) ^ { 2 }\), simplifying your answer.

2 Expand and simplify \(( r + 1 ) ^ { 4 } - r ^ { 4 }\). Use the method of differences together with the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$

1 Use the List of Formulae (MF10) to show that \(\sum _ { r = 1 } ^ { 13 } \left( 3 r ^ { 2 } - 5 r + 1 \right)\) and \(\sum _ { r = 0 } ^ { 9 } \left( r ^ { 3 } - 1 \right)\) have the same numerical value.

The positive variables \(y\) and \(t\) are related by $$y = a ^ { t }$$ where \(a\) is a positive constant.

1. (a) By differentiating \(\ln y\) with respect to \(t\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = a ^ { t } \ln a\).
   (b) Write down \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Determine the set of values of \(a\) for which the infinite series $$y + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} t ^ { 3 } } + \ldots$$ is convergent.
   A curve has parametric equations $$x = t ^ { a } , \quad y = a ^ { t }$$
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(a\) and \(t\), and show that, when \(t = 2\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 ^ { 1 - 2 a } ( 1 - a + 2 \ln a ) \ln a$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

Prove by induction that $$\sum _ { n = 1 } ^ { N } n ^ { 3 } = \frac { 1 } { 4 } N ^ { 2 } ( N + 1 ) ^ { 2 }$$ Use this result, together with the formula for \(\sum _ { n = 1 } ^ { N } n ^ { 2 }\), to show that $$\sum _ { n = 1 } ^ { N } \left( 20 n ^ { 3 } + 36 n ^ { 2 } \right) = N ( N + 1 ) ( N + 3 ) ( 5 N + 2 ) .$$ Let $$S _ { N } = \sum _ { n = 1 } ^ { N } \left( 20 n ^ { 3 } + 36 n ^ { 2 } + \mu n \right)$$ Find the value of the constant \(\mu\) such that \(S _ { N }\) is of the form \(N ^ { 2 } ( N + 1 ) ( a N + b )\), where the constants \(a\) and \(b\) are to be determined. Show that, for this value of \(\mu\), $$5 + \frac { 22 } { N } < N ^ { - 4 } S _ { N } < 5 + \frac { 23 } { N }$$ for all \(N \geqslant 18\).

4 Let \(\mathrm { f } ( r ) = r ( r + 1 ) ( r + 2 )\). Show that $$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = 3 r ( r + 1 )$$ Hence show that \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )\). Using the standard result for \(\sum _ { r = 1 } ^ { n } r\), deduce that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\). Find the sum of the series $$1 ^ { 2 } + 2 \times 2 ^ { 2 } + 3 ^ { 2 } + 2 \times 4 ^ { 2 } + 5 ^ { 2 } + 2 \times 6 ^ { 2 } + \ldots + 2 ( n - 1 ) ^ { 2 } + n ^ { 2 }$$ where \(n\) is odd.