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

5 Prove by induction that for all integers \(n \geqslant 1\) $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 1 \right) ( r ! ) = n ( n + 1 ) !$$

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1. Use the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = a n ( n + 1 ) ( n + b )$$ where \(a\) and \(b\) are constants to be determined.
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2. Hence, or otherwise, find a fully factorised expression for $$\sum _ { r = n + 1 } ^ { 5 n } r ( r + 3 )$$ $$\mathbf { A } = \left[ \begin{array} { c c } 3 & i - 1 \\ i & 2 \end{array} \right]$$

13 Prove by induction that, for all integers \(n \geq 1\) $$\sum _ { r = 1 } ^ { n } 2 ^ { - r } = 1 - 2 ^ { - n }$$ [4 marks]

5 Prove by induction that, for all integers \(n \geq 1\), $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \left\{ \frac { 1 } { 2 } n ( n + 1 ) \right\} ^ { 2 }$$ [4 marks]

1 In this question you must show detailed reasoning.
The quadratic equation \(x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find a quadratic equation with roots \(\alpha + \frac { 1 } { \beta }\) and \(\beta + \frac { 1 } { \alpha }\). Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).

1 Find an expression for \(1 \times 2 ^ { 2 } + 2 \times 3 ^ { 2 } + 3 \times 4 ^ { 2 } + \ldots + n ( n + 1 ) ^ { 2 }\) in terms of \(n\). Give your answer in fully factorised form.

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.

1 Using any standard results given in the List of Formulae (MF20), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r + 1 \right) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 2 \right)$$ for all positive integers \(n\).

1 The series \(S\) is given by \(S = \sum _ { r = 0 } ^ { N } ( N + r ) ^ { 2 }\).

1. Write out the first three terms and the last three terms of the series for \(S\).
2. Use the standard result \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to show that \(S = \frac { 1 } { 6 } N ( N + 1 ) ( a N + 1 )\) for some positive integer \(a\) to be determined.

1 Using standard summation results, show that \(\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } + r \right) \equiv \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 ) ^ { 2 }\).

1 Using standard results given in MF20, show that $$\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 2 r ^ { 2 } + 5 \right) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 2 \right) ( 3 n + 8 )$$

Answer only one of the following two alternatives. **EITHER** Show that \(\left(n + \frac{1}{2}\right)^3 - \left(n - \frac{1}{2}\right)^3 \equiv 3n^2 + \frac{1}{4}\). [1] Use this result to prove that \(\sum_{n=1}^N n^2 = \frac{1}{6}N(N + 1)(2N + 1)\). [2] The sums \(S\), \(T\) and \(U\) are defined as follows: \begin{align} S &= 1^2 + 2^2 + 3^2 + 4^2 + \ldots + (2N)^2 + (2N + 1)^2,
T &= 1^2 + 3^2 + 5^2 + 7^2 + \ldots + (2N - 1)^2 + (2N + 1)^2,
U &= 1^2 - 2^2 + 3^2 - 4^2 + \ldots - (2N)^2 + (2N + 1)^2. \end{align} Find and simplify expressions in terms of \(N\) for each of \(S\), \(T\) and \(U\). [5] Hence

1. describe the behaviour of \(\frac{S}{T}\) as \(N \to \infty\), [1]
2. prove that if \(\frac{S}{U}\) is an integer then \(\frac{T}{U}\) is an integer. [3]

**OR** The curves \(C_1\) and \(C_2\) have polar equations $$r = 4\cos\theta \quad \text{and} \quad r = 1 + \cos\theta$$ respectively, where \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Show that \(C_1\) and \(C_2\) meet at the points \(A\left(\frac{4}{3}, \alpha\right)\) and \(B\left(\frac{4}{3}, -\alpha\right)\), where \(\alpha\) is the acute angle such that \(\cos\alpha = \frac{1}{3}\). [2]
2. In a single diagram, draw sketch graphs of \(C_1\) and \(C_2\). [3]
3. Show that the area of the region bounded by the arcs \(OA\) and \(OB\) of \(C_1\), and the arc \(AB\) of \(C_2\), is $$4\pi - \frac{1}{3}\sqrt{2} - \frac{13}{2}\alpha.$$ [7]

Prove that $$\sum_{r=1}^{n} 6(r^2 - 1) = (n - 1)n(2n + 5).$$ [4]

Given that $$(2r + 1)^3 = Ar^3 + Br^2 + Cr + 1,$$

1. find the values of the constants \(A\), \(B\) and \(C\). [2]
2. Show that $$(2r + 1)^3 - (2r - 1)^3 = 24r^2 + 2.$$ [2]
3. Using the result in part (b) and the method of differences, show that $$\sum_{r=1}^n r^2 = \frac{1}{6}n(n + 1)(2n + 1).$$ [5]

Use the formulae for \(\sum_{r=1}^{n} r^3\) and \(\sum_{r=1}^{n} r^2\) to find the value of $$\sum_{r=3}^{60} r^2(r - 6)$$ [4 marks]

Use the standard results for \(\sum_{r=1}^n r\) and \(\sum_{r=1}^n r^2\) to show that, for all positive integers \(n\), $$\sum_{r=1}^n (6r^2 + 2r + 1) = n(2n^2 + 4n + 3).$$ [6]

Find \(\sum_{r=1}^{n} (r-1)(r+1)\), giving your answer in a fully factorised form. [6]

Find \(\sum_{r=1}^{n} (2r - 1)^2\), expressing your answer in a fully factorised form. [6]

Find $$\sum_{r=1}^{20}(r^2 - 2r)$$ Circle your answer. [1 mark] 2450 \quad 2660 \quad 5320 \quad 43680