Question 1: Prove by induction that Sum of 1/(i(i+ 1))=n/(n+ 1) Question 2: Prov

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Question 1: Prove by induction that Sum of 1/(i(i+ 1))=n/(n+ 1)
Question 2: Prove or disprove: the product of an irrational number by a rational number is always irrational 1. Show that ⌈n/2⌉*⌊n/2⌋=⌊n^2/4⌋ (hint: check all possible cases)
Question 1: Prove by induction that Sum of 1/(i(i+ 1))=n/(n+ 1)
Question 2: Prove or disprove: the product of an irrational number by a rational number is always irrational 1. Show that ⌈n/2⌉*⌊n/2⌋=⌊n^2/4⌋ (hint: check all possible cases)
Question 2: Prove or disprove: the product of an irrational number by a rational number is always irrational 1. Show that ⌈n/2⌉*⌊n/2⌋=⌊n^2/4⌋ (hint: check all possible cases)

Explanation / Answer

To prove by induction, first we check for n = 1, sum would be equal to 1/1*2 = 1/2 = RHS (n/n+1 , put n = 1). Hence it is true for n = 1. Now we assume for n = k. Sum = k/k+1 For n = k + 1, sum = k/k+1 + 1/(k+1)(k+2) = (k^2 + 2k + 1)/(k+1)(k+2) = k+1/k+2 . Hence proved. Product of an irrational number with a rational number is always rational. We can prove by contradiction. Assume product of irrational and rational number is rational, then we can write the irrational number as division of two rational number, but quotient of two rational should always be rational, hence contradiction

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