Standard summation formulae application

Prove by induction that, for \(n \geq 1\), \(\sum_{r=1}^{n} r(r + 1) = \frac{1}{3}n(n + 1)(n + 2)\). [5]

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

The first four terms of the series \(S\) can be written as $$S = (1 \times 2) + (2 \times 3) + (3 \times 4) + (4 \times 5) + ...$$

1. Write an expression, using \(\sum\) notation, for the sum of the first \(n\) terms of \(S\) [1 mark]
2. Show that the sum of the first \(n\) terms of \(S\) is equal to $$\frac{1}{3}n(n + 1)(n + 2)$$ [2 marks]

Find \(\sum_{r=1}^{n}(r+1)(r+5)\). Give your answer in a fully factorised form. [4]

Find an expression, in terms of \(n\), for the sum of the first \(n\) terms of the series $$1.2.4 + 2.3.5 + 3.4.6 + \ldots + n(n + 1)(n + 3) + \ldots$$ Express your answer as a product of linear factors. [6]

Using the formulae for \(\sum_{r=1}^{n} r\) and \(\sum_{r=1}^{n} r^2\), show that \(\sum_{r=1}^{10} r(3r - 2) = 1045\). [3]

Prove that $$\sum_{r=1}^{n} 18(r^2 - 4) = n(6n^2 + 9n - 69).$$ [4 marks]

Find \(\sum_{r=1}^{n}(2r^2 - 1)\), expressing your answer in fully factorised form. [4]

In this question you must show detailed reasoning. In this question you may assume the results for $$\sum_{r=1}^{n} r^3, \quad \sum_{r=1}^{n} r^2 \quad \text{and} \quad \sum_{r=1}^{n} r.$$ Show that the sum of the cubes of the first \(n\) positive odd numbers is $$n^2(2n^2 - 1).$$ [5]

Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series \((1 \times 2 \times 4) + (2 \times 3 \times 5) + (3 \times 4 \times 6) + \ldots\) where \(n\) is a positive integer. Give your answer in fully factorised form. [6]