



 
Hi Hossun. We need to give this a name so let f(n) = 1/(1x2)+1/(2x3)+1/(3x4)...+1/(n(n+1)). The first thing we notice is that for n > 1, we are just adding another fraction to the previous value of f(n). So we can construct f(n) = f(n1) + 1/(n(n+1)). Now look at the small values of n: So for the first few small values of n, we have proven by demonstration that f(n) = n / (n+1). Our task is to prove that if it works for any positive integer value of n, then it works for n + 1. This way, it must by induction work for all subsequent values of n. Formally said, we need to prove that if for some positive integer n we can show that f(n) = n / (n+1), then we can conclude that f(n+1) = (n + 1) / (n + 2). We begin the real "proof" by expanding f(n + 1): f(n + 1) = f(n) + 1 / ((n+1)((n+1)+1)) because that's based on the construction. = n / (n+1) + 1 / ((n+1)(n+2)) because f(n) = n / (n+1); this is called "using what you know from earlier". = n(n+2) / ((n+1)(n+2)) + 1 / ((n+1)(n+2)) because we can multiply the left fraction by (n+2)/(n+2). = (n^{2} + 2n + 1) / ((n+1)(n+2)) because we have a common denominator and can combine the numerators. = (n+1)^{2} / ( (n+1)(n+2)) because we can factor the numerator now; it is a perfect square. = (n+1) / (n+2) because we can cancel the common (n+1) factor from the numerator and denominator. Q.E.D. (which means "that which was to be proven", in other words: "voilà") Cheers,  


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