If you tutor math 12, you’ll have come across the premise
1 + 1/2 + 1/4 + 1/8 +….
Before we go any further, a reminder on how to multiply fractions:
(2/3) x (4/5) = 8/15. You just multiply the two top numbers and put that number on top. Then you multiply the two bottom ones and put that number on the bottom.
Now, back to 1 + 1/2 + 1/4 + 1/8 + 1/16+….(the terms continue forever)
To generate the next term, you just multiply the previous one by, in this case, 1/2.
(1/4) x (1/2) = 1/8
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 +….
In spite of the infinite number of terms, the answer is 2.
1 + 1/2 + 1/4 + 1/8 + 1/16 +1/32 + 1/64 + (the terms go on forever)….= 2.
How can there be a sum of an unending number of terms? Well, there is a proof for it – but that doesn’t mean you have to believe it. Perhaps the best answer is that from a mathematical point of view, you can add up an infinite number of terms. However, in this context, two things have to be true:
1) You must always multiply the current term by the same factor to get the next one.
2) That multiplying factor must be less than 1.
So there isn’t a sum to 1+1/2+1/3+1/4 + (the terms continue forever)….because the next number is not a constant multiple of the one before.
There isn’t a sum to 1 + 1 + 1 + (the terms continue forever)….because we’re multiplying the current term by exactly 1 to get the next term. We must multiply by the same number each time, but it must also be less than one.
There is a sum to 1 + 2/3 + 4/9 + 8/27 +….(infinite terms). Note that we are always multiplying by 2/3 to get the next term. The sum is 3.
The formula for the sum is
a is the first term
r is the multiplying factor to get the next term (called the common ratio)
If you use a calculator, be sure to put the brackets around the denominator as shown in the formula.
Thanks for stopping by.
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.