Deducing formula for nth term in sequence and validate using principles of induction

Question

I a working my way through some old exam papers but have come up with a problem. One question on sequences and induction goes:

A sequence of integers $x_1, x_2,\cdots, x_k,\cdots$ is defind recursively as follows: $x_1 = 2$ and $x_{k+1} = 5x_k,$ for $k \geq 1.$

i) calculate $x_2, x_3, x_4$

ii) deduce a formula for the $n$th term i.e. $x_n$ in terms of $n$ and then prove its validity, using the principles of mathematical induction.

It is the last part that is giving me trouble. I think $x_2, x_3$ and $x_4$ are $10, 50$ and $250$ respectively. I also think I managed to work out the formula, it is $f(n) = 2 \cdot 5^{n-1}.$

However I'm not sure how I'm supposed to prove this using induction... induction is only used when you're adding the numbers in a sequence I thought? I've looked everywhere and can't find any answer specific enough to this question to help. Any help appreciated. Thanks.

Answer 1

Your answer is correct.

To write down the proof by induction you have:

Your formula is correct for $n=1$ because $x_1=2\cdot 5^{1-1}=2\cdot 5^0=2.$

Now you suppose that it is correct for $n,$ that is, $x_n=2\cdot 5^{n-1},$ and you need to prove that it hols for $n+1.$ We have:

$x_{n+1}=5x_n= 5\cdot (2\cdot 5^{n-1})=2\cdot 5^n,$ (where we have used the induction hypothesis in the second equality) which finishes the proof.

Answer 2

Induction is aplicable in many cases, not just sums. In this case you have a expression for $x_n$, prove (base) that it is valid for $n = 1$ and (induction) if it is valid for $n$ it is valid for $n + 1$.