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Write as a sum of fractions 1/y(y+1)

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landishigh | Student, College Freshman | (Level 1) Honors

Posted December 4, 2010 at 1:53 AM via web

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Write as a sum of fractions 1/y(y+1)

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted December 4, 2010 at 1:54 AM (Answer #1)

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We have to express 1 / y(y+1) as the sum of two fractions.

Let us write 1 / y(y+1) as A / y + B / (y + 1)

=> [A( y + 1) + By]/ y(y+1) = 1 / y(y+1)

Cancel y(y+1)

=> Ay + A + By = 1

=> y ( A + B)  + A = 1

Now equate the terms with y and the numeric terms.

=> A + B = 0 and A = 1

=> A = 1 and B = -A = -1

Therefore we can write 1 / y(y+1) as 1/y - 1/(y+1)

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted December 4, 2010 at 1:53 AM (Answer #2)

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We remark that the denominator of the given ratio is the least common denominator of 2 irreducible ratios.

The final ratio 1/y(y+1) is the result of addition or subtraction of 2 elementary fractions, as it follows:

1/y(y+1) = A/y + B/(y+1) (1)

We'll multiply by y(y+1) both sides:

1 = A(y+1) + By

We'll remove the brackets:

1 = Ay + A + By

We'll factorize by y to the right side:

1 = y(A+B) + A

We'll compare expressions of both sides:

A+B = 0

A = 1

1 + B = 0

B = -1

We'll substitute A and B into the expression (1) and we'll get the algebraic sum of 2 elementary fractions:

1/y(y+1) = 1/y - 1/(y+1)

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