Prove summation with exponentials

2 Prove by mathematical induction that, for all positive integers \(n\), $$1 + 2 x + 3 x ^ { 2 } + \ldots + n x ^ { n - 1 } = \frac { 1 - ( n + 1 ) x ^ { n } + n x ^ { n + 1 } } { ( 1 - x ) ^ { 2 } }$$

1. (i) A sequence of positive numbers is defined by

$$\begin{aligned} u _ { 1 } & = 5 \\ u _ { n + 1 } & = 3 u _ { n } + 2 , \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 2 \times ( 3 ) ^ { n } - 1$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } \frac { 4 r } { 3 ^ { r } } = 3 - \frac { ( 3 + 2 n ) } { 3 ^ { n } }$$

9. (a) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n }$$ (b) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 0 , \quad u _ { 2 } = 32 , \\ u _ { n + 2 } = 6 u _ { n + 1 } - 8 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 4 ^ { n + 1 } - 2 ^ { n + 3 }$$

6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r 2 ^ { r - 1 } = 1 + ( n - 1 ) 2 ^ { n }\).

5 Prove by induction that, for \(n \geqslant 1 , \sum _ { r = 1 } ^ { n } 4 \times 3 ^ { r } = 6 \left( 3 ^ { n } - 1 \right)\).

7 Prove by induction that \(12 + 36 + 108 + \ldots + 4 \times 3 ^ { n } = 6 \left( 3 ^ { n } - 1 \right)\) for all positive integers \(n\).

6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r 3 ^ { r - 1 } = \frac { 1 } { 4 } \left[ 3 ^ { n } ( 2 n - 1 ) + 1 \right]\). Section B (36 marks)

8

1. Prove by mathematical induction that, for \(z \neq 1\) and all positive integers \(n\), $$1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = \frac { z ^ { n } - 1 } { z - 1 }$$
2. By letting \(z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to deduce that $$\sum _ { m = 1 } ^ { \infty } \left( \frac { 1 } { 2 } \right) ^ { m } \sin m \theta = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$

9 Prove by induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 5 - 4 r } { 5 ^ { r } } = \frac { n } { 5 ^ { n } }$$

6

1. Show that \(\frac { 1 } { ( k + 2 ) ! } - \frac { k + 1 } { ( k + 3 ) ! } = \frac { 2 } { ( k + 3 ) ! }\).
2. Prove by induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \frac { r \times 2 ^ { r } } { ( r + 2 ) ! } = 1 - \frac { 2 ^ { n + 1 } } { ( n + 2 ) ! }$$ (6 marks)
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3

1. Express \(( k + 1 ) ^ { 2 } + 5 ( k + 1 ) + 8\) in the form \(k ^ { 2 } + a k + b\), where \(a\) and \(b\) are constants.
2. Prove by induction that, for all integers \(n \geqslant 1\), $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) \left( \frac { 1 } { 2 } \right) ^ { r - 1 } = 16 - \left( n ^ { 2 } + 5 n + 8 \right) \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$$

5 Prove by induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)\).

5 Prove by induction that \(\sum _ { r = 1 } ^ { n } r \times 2 ^ { r - 1 } = 1 + ( n - 1 ) 2 ^ { n }\) for all positive integers \(n\).

4

1. Prove by induction that $$2 + ( 3 \times 2 ) + \left( 4 \times 2 ^ { 2 } \right) + \ldots + ( n + 1 ) 2 ^ { n - 1 } = n 2 ^ { n }$$ for all integers \(n \geqslant 1\).
2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n } \left( 2 ^ { n + 1 } - 1 \right)$$

6 Prove by induction that $$\frac { 2 \times 1 } { 2 \times 3 } + \frac { 2 ^ { 2 } \times 2 } { 3 \times 4 } + \frac { 2 ^ { 3 } \times 3 } { 4 \times 5 } + \ldots + \frac { 2 ^ { n } \times n } { ( n + 1 ) ( n + 2 ) } = \frac { 2 ^ { n + 1 } } { n + 2 } - 1$$ for all integers \(n \geqslant 1\).

6

1. Show that $$\left( 1 - \frac { 1 } { ( k + 1 ) ^ { 2 } } \right) \times \frac { k + 1 } { 2 k } = \frac { k + 2 } { 2 ( k + 1 ) }$$
2. Prove by induction that for all integers \(n \geqslant 2\) $$\left( 1 - \frac { 1 } { 2 ^ { 2 } } \right) \left( 1 - \frac { 1 } { 3 ^ { 2 } } \right) \left( 1 - \frac { 1 } { 4 ^ { 2 } } \right) \ldots \left( 1 - \frac { 1 } { n ^ { 2 } } \right) = \frac { n + 1 } { 2 n }$$

Prove by induction that, for \(n \in \mathbb{Z}^+\), \(\sum_{r=1}^{n} r 2^r = 2\{1 + (n - 1)2^n\}\). [5]

Prove by induction that \(3 + 6 + 12 + \ldots + 3 \times 2^{n-1} = 3(2^n - 1)\) for all positive integers \(n\). [7]

Prove by induction that \(\sum_{r=1}^{n} 3^{r-1} = \frac{3^n - 1}{2}\). [6]