Question
Let $$a_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}}$$ be a sequence with $n$ th term $a_{n}$.Find expressions for $a_{n+1}$ and $a_{n+2}$ in terms of $n$.
Step 1
By definition, we have $$a_{n+1}=\frac{(1+\sqrt{5})^{n+1}-(1-\sqrt{5})^{n+1}}{2^{n+1} \sqrt{5}}.$$ We can rewrite this as $$a_{n+1}=\frac{1}{2} \cdot \frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n} \sqrt{5}} + Show more…
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