<s>[INST] Question regarding proving by induction I am struggling with a math problem I have been assigned. The problem is as follows:
Let $X_1 = -3$ and $X_2 = 0$. Given that for every natural number $n \geq 2, X_{n+1} = 7X_n - 10X_{n-1}$, prove by induction that for every $n$ belonging to $\mathbb{N}$, $X_n = 2 \cdot 5^{n-1} - 5 \cdot 2^{n-1}$
Right now all I have proven so far is the two base cases, I am not sure how to take this proof any further. Any assistance you can offer is greatly appreciated, thank you.
 [/INST] Start with your recurrence relation and substitute your inductive assumption:
$$X_{n+1} = 7(2 \cdot 5^{n-1} - 5 \cdot 2^{n-1}) - 10(2 \cdot 5^{n-2} - 5 \cdot 2^{n-2}) \\
= 14 \cdot 5^{n-1} - 35 \cdot 2^{n-1} - 20 \cdot 5^{n-2} + 50 \cdot 2^{n-2} \\
= 10 \cdot 5^{n-1} - 10 \cdot 2^{n-1} \\
= 2 \cdot 5^n - 5 \cdot 2^n.$$
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