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Fill in the blanks to make a true equation:

(the question marks are the blanks, and the two question marks don't have to be the same number)

 

\(\frac{1}{n(n + 3)} = \frac{?}{n} - \frac{?}{n + 3}.\)

 May 5, 2023
 #1
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Let the constants be A and B, so 1/(n(n + 3)) = A/n + B/(n + 3).

 

We can solve for A and B by multiplying both sides of the equation by n(n + 3), then combining like terms. We get:

A(n + 3) + Bn = 1

We can solve for A and B by setting n = 0 and n = -3, respectively. When n = 0, the left-hand side of the equation is 0, so we have:

A(3) + B(0) = 1

This gives us A = 1/3. When n = -3, the left-hand side of the equation is 1, so we have:

A(-2) + B(-3) = 1

This gives us B = -1/6. Therefore, A = 1/3 and B = -1/6, and we have:

1/(n(n + 3)) = 1/3n - 1/6(n + 3)

 May 5, 2023

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