resolve 1/r(r+1) into partial fraction
1/(r(r+1)) = A/r + B/(r+1)
A(r+1) + Br = 1
(A+B)r + A = 1
so A = 1
A+B = 0 ==> B = -1
1/(r(r+1)) = 1/r - 1/(r+1)
To resolve the expression 1/r(r+1) into partial fractions, follow these steps:
Step 1: Factorize the denominator.
The denominator is r(r+1), which is already factored.
Step 2: Write the partial fraction form.
1/r(r+1) can be expressed as A/r + B/(r+1), where A and B are constants.
Step 3: Find the common denominator.
The common denominator of A/r and B/(r+1) is r(r+1).
Step 4: Combine the two fractions.
Multiply the A term by (r+1) and the B term by r to get a common denominator:
(A(r+1) + Br)/(r(r+1))
Step 5: Equate the numerators.
Equate the numerators and set them equal to the original numerator, which is 1:
A(r+1) + Br = 1
Step 6: Simplify the equation.
Expand the equation and group like terms:
Ar + A + Br = 1
Step 7: Solve for A and B.
Separate the terms with r and the constants:
(A + B)r + A = 1
Equate the coefficients of r and the constants:
A + B = 0 (1)
A = 1 (2)
Solve the system of equations:
From equation (1), we have B = -A = -1.
From equation (2), we have A = 1.
Therefore, A = 1 and B = -1.
Step 8: Substitute the values of A and B back into the partial fraction form.
1/r(r+1) = 1/r - 1/(r+1)
Therefore, the expression 1/r(r+1) can be resolved into the partial fractions as 1/r - 1/(r+1).
To resolve the expression 1/(r(r+1)) into partial fractions, follow these steps:
Step 1: Factorize the denominator.
In this case, the denominator is r(r+1). It is already factorized.
Step 2: Write the expression as a sum of fractions.
1/(r(r+1)) can be written as A/r + B/(r+1), where A and B are constants that we need to determine.
Step 3: Find the common denominator.
The common denominator for the two fractions is r(r+1).
Step 4: Set up the equation.
Write the original expression as a single fraction with the common denominator:
1/(r(r+1)) = A/r + B/(r+1)
Step 5: Clear the fractions.
Multiply both sides of the equation by the common denominator (r(r+1)) to clear the fractions:
1 = A(r+1) + Br
Simplify the equation:
1 = Ar + A + Br
Step 6: Solve for A and B.
Compare the coefficients of r on both sides of the equation:
0r = (A + B)r
This means that for the equation to hold true, A + B must be equal to zero. Therefore, A = -B.
Compare the constant terms on both sides of the equation:
1 = A
Therefore, A is equal to 1.
Since A = -B, we can conclude that B is equal to -1.
Therefore, the partial fraction decomposition of 1/(r(r+1)) is:
1/r - 1/(r+1)