Prove summation formula

1. (a) Prove that the sum of the first \(n\) positive integers is given by

$$\frac { 1 } { 2 } n ( n + 1 ) .$$ (b) Hence, find the sum of

1. the integers from 100 to 200 inclusive,
2. the integers between 300 to 600 inclusive which are divisible by 3 .

6

1. Show that $$( k + 1 ) \left( 4 ( k + 1 ) ^ { 2 } - 1 \right) = 4 k ^ { 3 } + 12 k ^ { 2 } + 11 k + 3$$
2. Prove by induction that, for all integers \(n \geqslant 1\), $$1 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } + \ldots + ( 2 n - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$

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

1. (a) Prove by induction that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( 2 r + 1 ) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n + 2 )$$ (b) Hence, show that, for all positive integers \(n\), $$\sum _ { r = n } ^ { 2 n } r ( r + 1 ) ( 2 r + 1 ) = \frac { 1 } { 2 } n ( n + 1 ) ( a n + b ) ( c n + d )$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be determined.

1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)

$$\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r + 2 ) = n ( n + 2 ) ( n + 3 )$$ (ii) Prove by induction that for all positive odd integers \(n\) $$f ( n ) = 4 ^ { n } + 5 ^ { n } + 6 ^ { n }$$ is divisible by 15

1. Prove by induction that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$

5. Prove that $$\sum _ { r = 1 } ^ { n } 6 \left( r ^ { 2 } - 1 \right) \equiv ( n - 1 ) n ( 2 n + 5 )$$

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

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]