Find the sum of the geometric series thirteen halves plus thirteen fourths plus thirteen eighths.
The formula for a sum that’s going to infinity is 𝑎 divided by one minus 𝑟, where 𝑎 is the initial amount, and in this case that will be thirteen halves, and 𝑟 is the common ratio. The common ratio is the number that we multiply to each term to get the following term.
So we can solve for 𝑟 by dividing. If we would take thirteen fourths and divide by thirteen halves, that would tell us what we multiply thirteen halves by in order to get thirteen fourths. However, when we divide fractions, we multiply by the reciprocal. So instead of dividing, we multiply. And we multiply by the reciprocal, so we flip our second fraction. We can multiply straight across and get twenty-six fifty seconds and then reduce or cancel the thirteenths and two goes into four twice. So on the numerator, there’s nothing, so that means there’s really a one. And on the denominator, there’s just a two. So 𝑟 is equal to one-half, and now we can play again.
So we have thirteen halves divided by one minus one-half. So let’s first begin with the denominator. One minus one-half is just one-half now. Now, that’s a pretty easy fraction to work with, but just a kind of review, when we add and subtract fractions, we need common denominators. So we could change one into two over two and we subtract the numerators and keep our denominator, which again is one-half.
Two minus one is one and then we keep the two on the bottom, so thirteen halves divided by one-half, which we could rewrite like this. And then remember we flip and multiply. The twos cancel. So this means the sum of our geometric series is 13.