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Level 2

Let A = 2 1 3 + 2 3 5 + 2 5 7 + 2 7 9 + 2 9 11 A=\dfrac{2}{1*3}+\dfrac{2}{3*5}+\dfrac{2}{5*7}+\dfrac{2}{7*9}+\dfrac{2}{9*11} and let B = 1 1 2 + 1 2 3 + 1 3 4 + 1 4 5 + 1 5 6 + 1 6 7 + 1 7 8 B=\dfrac{1}{1*2}+\dfrac{1}{2*3}+\dfrac{1}{3*4}+\dfrac{1}{4*5}+\dfrac{1}{5*6}+\dfrac{1}{6*7}+\dfrac{1}{7*8} ,then evaluate 33 A 32 B 33A-32B .


The answer is 2.

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1 solution

Ewerton Cassiano
Jan 20, 2014

Using the theory of telescopic sum, we have 1 k ( k + n ) \frac{1}{k(k+n)} , where we have: 1 k \frac{1}{k} - 1 k + n \frac{1}{k + n} , but if n > 1 n > 1 , we will need to divide the numerator by n n . Starting: For A A we have: 2 k ( k + 2 ) \frac{2}{k(k+2)} = 2 1 \frac{2}{1} - 2 3 \frac{2}{3} + 2 3 \frac{2}{ 3 } + ... + 2 9 \frac{2}{9} - 2 11 \frac{2}{11} . Soon , we have : 2 2 - 2 11 \frac{2}{11} = 20 11 \frac{20}{11} , dividing the numerator by n n , we have: 10 11 \boxed{\frac{10}{11}} . For B B we have: 1 1 \frac{1}{1} - 1 2 \frac{1}{2} + 1 2 \frac{1}{2} - 1 3 \frac{1}{3} + ... + 1 7 \frac{1}{7} - 1 8 \frac{1}{8} . So, 1 1 - 1 8 \frac{1}{8} = 7 8 \boxed{\frac{7}{8}} . Applying 33 A 32 B 33A - 32B = 33 10 11 33*\frac{10}{11} - 32 7 8 32*\frac{7}{8} = 30 28 30 - 28 = 2 \boxed{2}

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