How would I algebraically write this simple exponential word problem?

Question

I'm just a high-school student who's bad at math but truly does have an interest in it, I just have a hard time fully understanding things and remembering them.

So I have to write a function for this really simple exponential word problem. I won't write the whole problem out, instead, I'll just share the part I need help with.

Basically I just need to find a function that describes this sequence:

$(5 \times 2)2)2)2)2)2)2)...$

f(x), x being the number of times the initial value (5) is doubled.

I almost can't believe I can't figure this out. Maybe I'm just tired

Answer 1

It is called a geometric progression (GP). The function that describes the given sequence is the general term of this GP. $$f(x)=5(2)^{x-1},\,x\in\mathbb N. $$

Answer 2

There are several ways to get the answer to this problem. But let's first rewrite the equation in a some condensed form. $$y = 5 * 2^n, n \in \mathbb{N_0}$$ where n represents the number of time up to which sequence is repeated.

• First Method:

Just directly substitute the value of $n$ in the equation and get the answer.

• Second Method:

Take natural $log_2$ both side in the equation. It's a monotonic function hence we can do this operation. $$log_2(y) = log_2(5) + n log_2(2)$$ $$log_2(y) = log_2(5) + n$$ Tae the inverse of logarithm function both side. $$y = 2^{log_2(5) + n}$$