I was helping my daughter with her physics homework. The problem had a bee flying back and forth between two bikes moving toward each other.
The question was to see how far the bee traveled. Turned out that the first trip was 15km. The second was 7.5km. The third was 3.75km and the fourth was 1.875. (We estimated that the final answer was 30km)
I realized that each trip was half as far as the last one. But the total was not possible to calculate because there would technically be infinite trips.
As I tried to think of a way to express this mathematically, it reminded me of my long ago calculus classes. (I don’t use calculus in my day job, so I have mostly forgotten it.)
I am done with what my daughter needs to show for her problem, but I am curious:
How do you express and solve dividing a number by two and then repeatedly taking the quotient and dividing by two, getting the sum of all those numbers?
For example:
15 + 15/2 + 7.5/2 + 3.75/2 + 1.875/2 + ...... + 15/2^(n-1)=30
The first thing to recognize is that repeatedly dividing by 2 is the same as repeatedly multiplying by 1/2. This seemingly irrelevant change is important because you'll find more information about repeated multiplication, simply because we prefer to talk about multiplication instead of division when possible.
Finding the sum of repeated multiplications is known as finding the sum of a geometric series; the values you computed are known as its terms. The sum of finitely many terms (starting from the first and adding each of them until some number of them have been included) is known as a partial sum. We can even talk about the sum of the whole series, which is the limit of the partial sums viewed as a sequence.
This should provide some search terms if you want to know more, but here's what you'd be likely to find in, say, a high school algebra class: a formula for any partial sum and a formula for the sum. Call the first term $a$ (for your problem, $15$) and let $r$ be such that the second term is $ar$ (i.e., let $r$ be the ratio between terms; and for your problem, $1/2$). Assuming $r \ne 1$, then the partial sum of the first $n$ terms is $a\left(\dfrac{1-r^n}{1-r}\right)$. If and only if $|r| < 1$, then the sum exists and is $\dfrac{a}{1-r}$.