Prove divisibility

1 Prove by mathematical induction that \(2 ^ { 4 n } + 31 ^ { n } - 2\) is divisible by 15 for all positive integers \(n\).

1 Prove by mathematical induction that, for all positive integers \(n , 5 ^ { 3 n } + 32 ^ { n } - 33\) is divisible by 31 .

2 Prove by mathematical induction that \(6 ^ { 4 n } + 38 ^ { n } - 2\) is divisible by 74 for all positive integers \(n\).

2 Prove by mathematical induction that \(7 ^ { 2 n } - 1\) is divisible by 12 for every positive integer \(n\).

2 Prove by mathematical induction that, for all positive integers \(n , 7 ^ { 2 n } + 97 ^ { n } - 50\) is divisible by 48. [6]

2 It is g \(n\)th \(\mathrm { t } \phi ( n ) = 5 ^ { n } ( 4 n + 1 )\), ff \(\quad \mathbf { o } \quad n = , \mathbb { B } .\).
Pro tyn ath matical id tin th \(t \phi ( n )\) is \(\dot { \mathbf { d } } \dot { \mathbf { v } }\) sib eff \(\quad \mathbf { o }\) eyp itie in eg \(r n\).

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

$$f ( n ) = 7 ^ { n } - 2 ^ { n } \text { is divisible by } 5$$

9. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21

1. Prove, by induction, that for \(n \in \mathbb { Z } , n \geqslant 2\)

$$4 ^ { n } + 6 n - 10$$ is divisible by 18

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

$$f ( n ) = 7 ^ { n - 1 } + 8 ^ { 2 n + 1 }$$ is divisible by 57
(6)

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

$$\mathrm { f } ( n ) = 5 ^ { n } + 8 n + 3 \text { is divisible by } 4$$

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

1. \(\mathrm { f } ( n ) = 5 ^ { n } + 8 n + 3\) is divisible by 4 ,
2. \(\left( \begin{array} { l l } 3 & - 2 \\ 2 & - 1 \end{array} \right) ^ { n } = \left( \begin{array} { l r } 2 n + 1 & - 2 n \\ 2 n & 1 - 2 n \end{array} \right)\)

7. $$f ( n ) = 2 ^ { n } + 6 ^ { n }$$

1. Show that \(\mathrm { f } ( k + 1 ) = 6 \mathrm { f } ( k ) - 4 \left( 2 ^ { k } \right)\).
2. Hence, or otherwise, prove by induction that, for \(n \in \mathbb { Z } ^ { + } , \mathrm { f } ( n )\) is divisible by 8 .

10. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$f ( n ) = 2 ^ { 2 n - 1 } + 3 ^ { 2 n - 1 } \text { is divisible by } 5 .$$

5. Prove, by induction, that \(3 ^ { 2 n } + 7\) is divisible by 8 for all positive integers \(n\).

9. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$f ( n ) = 8 ^ { n } - 2 ^ { n }$$ is divisible by 6

1. Prove by induction that

$$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$ is divisible by 7 for all positive integers \(n\).

8. Prove by induction that \(4 ^ { n + 2 } + 5 ^ { 2 n + 1 }\) is divisible by 21 for all positive integers \(n\).

2 Prove by mathematical induction that \(5 ^ { 2 n } - 1\) is divisible by 8 for every positive integer \(n\).

3 Prove by mathematical induction that, for all non-negative integers \(n\), $$11 ^ { 2 n } + 25 ^ { n } + 22$$ is divisible by 24 .

3
- 2
0 \end{array} \right) .$$ Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\). Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\). 2 Show that the difference between the squares of consecutive integers is an odd integer. Find the sum to \(n\) terms of the series $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots + \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } + \ldots$$ and deduce the sum to infinity of the series. 3 It is given that \(\phi ( n ) = 5 ^ { n } ( 4 n + 1 ) - 1\), for \(n = 1,2,3 , \ldots\). Prove, by mathematical induction, that \(\phi ( n )\) is divisible by 8 , for every positive integer \(n\).

3 Prove by mathematical induction that, for all positive integers \(n , 10 ^ { n } + 3 \times 4 ^ { n + 2 } + 5\) is divisible by 9 .

2 Prove, by mathematical induction, that \(5 ^ { n } + 3\) is divisible by 4 for all non-negative integers \(n\).

2 It is given that \(\mathrm { f } ( n ) = 2 ^ { 3 n } + 8 ^ { n - 1 }\). By simplifying \(\mathrm { f } ( k ) + \mathrm { f } ( k + 1 )\), or otherwise, prove by mathematical induction that \(\mathrm { f } ( n )\) is divisible by 9 for every positive integer \(n\).

1 Prove by mathematical induction that \(3 ^ { 3 n } - 1\) is divisible by 13 for every positive integer \(n\).