Use mathematical induction to prove that the statement is true for every positive integer n. (1-1/2) (1-1/3) ...(1-1/n+1) = 1/n+1
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ANSWERS: 2
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Base case, put n=1 (1-(1/2)) = 1/(n+1) works. Now assume the equation is true for values of n up to N. If you can use those facts to prove it for N+1 then that proves it for all N. The N+1 example looks like: (1-(1/2))(1 - (1/3))...(1 - (1/(N+1)) (1 - 1/(N+2)) = 1/(N+2) According to the equation in the question, which we assume to be true when n=N, we know what the first N factors of that product come to: it's 1/(N+1) Replace the first N factors with 1/(N+1) and do some simplification and Bob's your uncle.
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Show(1-1/n+1) = 1/n+1. First use an easy value of n. Try n = 1. (1-1/1+1) = 1/1+1 (1 - 1/2) = (1/2) It works Now show it works with n=(n+1). Set the first term evaluated at n+1 equal to the second term evaluated at n added to itself evaluated at n+1.(Wherever the text used summation notation, you should, too, just in this first line.) 1 - 1/(n+1)+1 = 1/ n+1 + 1/(n+1)+1 The next step is the induction step. We proved already that (1-1/n+1) = 1/n+1. So replace one for the other. 1 - 1/(n+1)+1 = (1 - 1/n+1) + 1/(n+1)+1 From here on, just work the algebra on the right hand side of the equal sign until it looks like the left hand side. It will work out, if you do it right.
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