Simplify
(1-1/3)(1-1/3)(1-1/4)(1-1/5) . . . (1-2013)
I really need to understand this. I don't care about the answer. XD
just simplify and multiply
(1-1/2)(1-1/3)(1-1/4) = 1/2 * 2/3 * 3/4 = 1/4
You can see how they collapse, all except the last factor.
So, the long product above is just 1/2013
Try to do everything inside each individual parenthesis.
So in this case I see that there are a lot of fractions, try to convert everything into fractions and that will make the math inside the parenthesis much easier. Once you have them all you can go ahead and multiply them all.
To simplify the given expression, we can start by expanding it. Let's do it step by step:
(1 - 1/3)(1 - 1/3) = (2/3)(2/3) = 4/9
Now, let's multiply this result by the next fraction:
(4/9)(1 - 1/4) = (4/9)(3/4) = 12/36 = 1/3
We continue this process for all the remaining fractions until we reach (1 - 1/2013). Let's summarize the steps:
(1 - 1/3)(1 - 1/3)(1 - 1/4)(1 - 1/5)...(1 - 1/2013)
(2/3)(2/3)(1 - 1/4)(1 - 1/5)...(1 - 1/2013)
(4/9)(1 - 1/4)(1 - 1/5)...(1 - 1/2013)
(4/9)(3/4)(1 - 1/5)...(1 - 1/2013)
(12/36)(1 - 1/5)...(1 - 1/2013)
(12/36)(4/5)(1 - 1/6)...(1 - 1/2013)
Continuing this pattern, when we reach the last fraction (1 - 1/2013), we will have:
(12/36)(4/5)(5/6)(6/7)...(2012/2013)(1 - 1/2013)
Now, notice that the numerators of each fraction (12, 4, 5, 6, ..., 2012) form an arithmetic sequence with a common difference of 1. Similarly, the denominators (36, 5, 6, 7, ..., 2013) also form an arithmetic sequence with a common difference of 1.
So, we can use the formula for the sum of an arithmetic series to find the product of these fractions.
The formula for the sum of an arithmetic series is:
Sum = (n/2)(first term + last term)
In this case, the first term is 12, the last term is 2012, and the common difference is 1. The number of terms (n) is 2012 - 12 + 1 = 2001.
Using the formula, we can find:
Sum = (2001/2)(12 + 2012) = 1001 * 2024 = 2,029,024
Therefore, the simplified expression is 2,029,024.