Prove divisibility

4 Prove by mathematical induction that, for all positive integers \(n , 10 ^ { 3 n } + 13 ^ { n + 1 }\) is divisible by 7 .

4 Prove by mathematical induction that, for all non-negative integers \(n , 7 ^ { 2 n + 1 } + 5 ^ { n + 3 }\) is divisible by 44 .

7 Prove by induction that \(2 ^ { n + 1 } + 5 \times 9 ^ { n }\) is divisible by 7 for all integers \(n \geqslant 1\).

8 Prove by induction that \(11 \times 7 ^ { n } - 13 ^ { n } - 1\) is divisible by 3 , for all integers \(n \geqslant 0\).

7

1. Given that $$\mathrm { f } ( k ) = 12 ^ { k } + 2 \times 5 ^ { k - 1 }$$ show that $$\mathrm { f } ( k + 1 ) - 5 \mathrm { f } ( k ) = a \times 12 ^ { k }$$ where \(a\) is an integer.
2. Prove by induction that \(12 ^ { n } + 2 \times 5 ^ { n - 1 }\) is divisible by 7 for all integers \(n \geqslant 1\).

9 Prove by induction that \(5 ^ { n } + 2 \times 11 ^ { n }\) is divisible by 3 for all positive integers \(n\).

8 Prove by mathematical induction that \(8 ^ { n } - 3 ^ { n }\) is divisible by 5 for all positive integers \(n\).

4. Prove, by mathematical induction, that \(9 ^ { n } + 15\) is a multiple of 8 for all positive integers \(n\).

7. Prove, by mathematical induction, that \(13 ^ { ( 2 n - 1 ) } + 8\) is a multiple of 7 for all positive integers \(n\).
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1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\)

$$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$ is divisible by 7

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

$$f ( n ) = 3 ^ { 2 n + 4 } - 2 ^ { 2 n }$$ is divisible by 5
(6)

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

$$f ( n ) = 2 ^ { 3 n + 1 } + 3 \left( 5 ^ { 2 n + 1 } \right)$$ is divisible by 17

6 Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4 .

6

1. The function f is given by $$\mathrm { f } ( n ) = 15 ^ { n } - 8 ^ { n - 2 }$$ Express $$\mathrm { f } ( n + 1 ) - 8 \mathrm { f } ( n )$$ in the form \(k \times 15 ^ { n }\).
2. Prove by induction that \(15 ^ { n } - 8 ^ { n - 2 }\) is a multiple of 7 for all integers \(n \geqslant 2\).

4 The expression \(\mathrm { f } ( n )\) is given by \(\mathrm { f } ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }\).

1. Show that \(\mathrm { f } ( k + 1 ) - 16 \mathrm { f } ( k )\) can be expressed in the form \(A \times 3 ^ { 3 k }\), where \(A\) is an integer.
2. Prove by induction that \(\mathrm { f } ( n )\) is a multiple of 11 for all integers \(n \geqslant 1\).

12 Prove by induction that, for all \(n \in \mathbb { N }\), the expression $$5 ^ { n } - 2 ^ { n }$$ is divisible by 3
[0pt] [4 marks]
LL

6 Prove by induction that \(4 \times 8 ^ { n } + 66\) is divisible by 14 for all integers \(n \geqslant 0\).

7 Prove that \(2 ^ { 3 n } - 3 ^ { n }\) is divisible by 5 for all integers \(n \geqslant 1\).

3 Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4.

13 Define the repunit number, \(R _ { n }\), to be the positive integer which consists of a string of \(n 1\) 's. Thus, $$R _ { 1 } = 1 , \quad R _ { 2 } = 11 , \quad R _ { 3 } = 111 , \quad \ldots , \quad R _ { 7 } = 1111111 , \quad \ldots , \text { etc. }$$ Use induction to prove that, for all integers \(n \geqslant 5\), the number $$13579 \times R _ { n }$$ contains a string of ( \(n - 4\) ) consecutive 7's.

7 Let \(\mathrm { f } ( n ) = 11 ^ { 2 n - 1 } + 7 \times 4 ^ { n }\). Prove by induction that \(\mathrm { f } ( n )\) is divisible by 13 for all positive integers \(n\).

For \(n \in \mathbb{Z}^+\) prove that

1. \(2^{3n + 2} + 5^{n + 1}\) is divisible by 3, [9]
2. \(\begin{pmatrix} -2 & -1 \\ 9 & 4 \end{pmatrix}^n = \begin{pmatrix} 1-3n & -n \\ 9n & 3n+1 \end{pmatrix}\). [7]

$$f(n) = (2n + 1)7^n - 1.$$ Prove by induction that, for all positive integers \(n\), \(f(n)\) is divisible by 4. [6]

The polynomial \(\text{p}(n)\) is given by \(\text{p}(n) = (n-1)^3 + n^3 + (n+1)^3\).

1. 1. Show that \(\text{p}(k+1) - \text{p}(k)\), where \(k\) is a positive integer, is a multiple of 9. [3 marks]
   2. Prove by induction that \(\text{p}(n)\) is a multiple of 9 for all integers \(n \geqslant 1\). [4 marks]
2. Using the result from part (a)(ii), show that \(n(n^2 + 2)\) is a multiple of 3 for any positive integer \(n\). [2 marks]

Prove by induction that, for all integers \(n \geq 1\), the expression \(7^n - 3^n\) is divisible by 4 [4 marks]