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(b-a)/ab = 1/a – 1/b. What is the value of 1/2 + 1/6 + 1/1

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(b-a)/ab = 1/a – 1/b. What is the value of 1/2 + 1/6 + 1/1

by Max@Math Revolution » Mon Feb 25, 2019 3:59 pm

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(b-a)/ab = 1/a - 1/b. What is the value of 1/2 + 1/6 + 1/12 + ... + 1/90?

A. 4/5
B. 5/6
C. 8/9
D. 9/10
E. 11/12
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by Max@Math Revolution » Wed Feb 27, 2019 4:05 am
1/2 + 1/6 + 1/12 + ... + 1/90
= 1/(1*2) + 1/(2*3) + 1/(3*4) + ... + 1(9*10)
= (2-1)/(1*2) + (3-2)/(2*3) + (4-3)/(3*4) + ... + (10-9)/(9*10)
= (1/1 - 1/2) + (1/2 - 1/3) + (1/3-1/4) + ... + (1/9 - 1/10)
= 1/1 - 1/10
= 9/10

Therefore, D is the answer.
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by Scott@TargetTestPrep » Sat Mar 02, 2019 8:02 am
Max@Math Revolution wrote:(b-a)/ab = 1/a - 1/b. What is the value of 1/2 + 1/6 + 1/12 + ... + 1/90?

A. 4/5
B. 5/6
C. 8/9
D. 9/10
E. 11/12
1/2 = 1 - 1/2

1/6 = 1/2 - 1/3

1/12 = 1/3 - 1/4

...

1/90 = 1/9 - 1/10

Adding the above equations, we have:

1/2 + 1/6 + 1/12 + ... + 1/90 = 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/9 - 1/10

We see all the terms on the right hand side cancel except 1 and -1/10; therefore,

1/2 + 1/6 + 1/12 + ... + 1/90 = 1 - 1/10

1/2 + 1/6 + 1/12 + ... + 1/90 = 9/10

Answer: D

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