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(a) Prove by induction that $8^n - 1$ is divisible by 7 for all $n \in \mathbb{N}$ - Leaving Cert Mathematics - Question 4 - 2016

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(a) Prove by induction that $8^n - 1$ is divisible by 7 for all $n \in \mathbb{N}$. (b) Given $\log_2 2 = p$ and $\log_3 3 = q$, where $\alpha > 0$, write each of ... show full transcript

Worked Solution & Example Answer:(a) Prove by induction that $8^n - 1$ is divisible by 7 for all $n \in \mathbb{N}$ - Leaving Cert Mathematics - Question 4 - 2016

Step 1

Express $\log_8 \frac{8}{3}$ in terms of $p$ and $q$

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Answer

$\log_8 \frac{8}{3} = \log_8 8 - \log_8 3 = 1 - \log_8 3 = 1 - \frac{\log_a 3}{\log_a 8} = 1 - \frac{q}{p + 1}.$

Step 2

Express $\log_8 \frac{q^2}{16}$ in terms of $p$ and $q$

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Answer

$\log_8 \frac{q^2}{16} = \log_8 q^2 - \log_8 16 = 2\log_8 q - \log_8 16 = 2\frac{q}{p + 1} - 4 = \frac{2q}{p + 1} - 4.$

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