Prove sequence property via recurrence

6 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = n ^ { 2 } + 3 n\), for all positive integers \(n\).

1. Show that \(u _ { n + 1 } - u _ { n } = 2 n + 4\).
2. Hence prove by induction that each term of the sequence is divisible by 2 .

4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 3 ^ { 2 n } - 1$$

1. Write down the value of \(u _ { 1 }\).
2. Show that \(u _ { n + 1 } - u _ { n } = 8 \times 3 ^ { 2 n }\).
3. Hence prove by induction that each term of the sequence is a multiple of 8 .

7 It is given that \(u _ { n } = 13 ^ { n } + 6 ^ { n - 1 }\), where \(n\) is a positive integer.

1. Show that \(u _ { n } + u _ { n + 1 } = 14 \times 13 ^ { n } + 7 \times 6 ^ { n - 1 }\).
2. Prove by induction that \(u _ { n }\) is a multiple of 7 .

10 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 5 ^ { n } + 2 ^ { n - 1 }\).

1. Find \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Hence suggest a positive integer, other than 1 , which divides exactly into every term of the sequence.
3. By considering \(u _ { n + 1 } + u _ { n }\), prove by induction that your suggestion in part (ii) is correct. \section*{OCR}

4 It is given that \(\mathrm { f } ( n ) = 3 ^ { 3 n } + 6 ^ { n - 1 }\).

1. Show that \(\mathrm { f } ( n + 1 ) + \mathrm { f } ( n ) = 28 \left( 3 ^ { 3 n } \right) + 7 \left( 6 ^ { n - 1 } \right)\).
2. Hence, or otherwise, prove by mathematical induction that \(\mathrm { f } ( n )\) is divisible by 7 for every positive integer \(n\).

3 It is given that, for \(n = 0,1,2,3 , \ldots\), $$a _ { n } = 17 ^ { 2 n } + 3 ( 9 ) ^ { n } + 20$$ Simplify \(a _ { n + 1 } - a _ { n }\), and hence prove by induction that \(a _ { n }\) is divisible by 24 for all \(n \geqslant 0\).

4 The first five terms of the Fibonacci sequence, \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\), where \(n \geqslant 1\), are \(F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , F _ { 4 } = 3\) and \(F _ { 5 } = 5\).

1. Use the recurrence definition of the Fibonacci sequence, \(\mathrm { F } _ { \mathrm { n } + 1 } = \mathrm { F } _ { \mathrm { n } } + \mathrm { F } _ { \mathrm { n } - 1 }\), to express \(\mathrm { F } _ { \mathrm { n } + 4 }\) in terms of \(\mathrm { F } _ { \mathrm { n } }\) and \(\mathrm { F } _ { \mathrm { n } - 1 }\).
2. Hence prove by induction that \(\mathrm { F } _ { \mathrm { n } }\) is a multiple of 3 when \(n\) is a multiple of 4 .

3 For positive integers \(n\), the sequence of Fibonacci numbers, \(\left\{ \mathrm { F } _ { \mathrm { n } } \right\}\), starts with the terms \(F _ { 1 } = 1 , F _ { 2 } = 1 , F _ { 3 } = 2 , \ldots\) and is given by the recurrence relation \(\mathrm { F } _ { \mathrm { n } } = \mathrm { F } _ { \mathrm { n } - 1 } + \mathrm { F } _ { \mathrm { n } - 2 } ( \mathrm { n } \geqslant 3 )\).

1. Show that \(\mathrm { F } _ { 3 \mathrm { k } + 3 } = 2 \mathrm {~F} _ { 3 \mathrm { k } + 1 } + \mathrm { F } _ { 3 \mathrm { k } }\), where \(k\) is a positive integer.
2. Prove by induction that \(\mathrm { F } _ { 3 n }\) is even for all positive integers \(n\).