Give recursive definition of sequence $a_n = 2^n, n=2,3, 4... where $ $a_1 = 2$

Question

Give recursive definition of sequence $a_n = 2^n, n = 2, 3, 4... where $ $a_1 = 2$

I'm just not sure how to approach these problems.

Then it asks to give a def for:

$a_n = n^2-3n, n = 0, 1, 2...$

Thanks for all the help!

Answer 1

For the first one, write the term $a_{n+1}$ and compare it to $a_n$: $$a_{n+1}=2^{n+1}=2\cdot2^n=2a_n$$

For the second one, repeat the process: $$\begin{align}a_{n+1}&=(n+1)^2-3(n+1)\\ &=n^2+2n+1-3n-3\\ &=n^2-3n+2n-2\\ &=a_n+2(n-1) \end{align}$$

Answer 2

Note that there are lots of answers to these questions. IMO those given by Terra Hyde are probably what your instructor is expecting, but you could also say:

for the first one, $$a_{n+1}=a_n+2^n$$ and for the second $$a_{n+1}=a_n+a_{n-1}-n^2+7n-6$$ among many, many other possibilities.