Solve for A and B if:
1 A B
----- = --- + ---
k(k+1) k k+1
the 1 is over k(k+1), and the A is over k, and the B is over k+1
common denominator: k(k+1)
1/[(k)(k+1)]=A(k+1)/[(k)(k+1)] + B(k)/[(k)(k+1)]
1/[(k)(k+1)]= [ A(k+1)+ B(k)]/ [(k)(k+1)]
cross multiply:
[(k)(k+1)]= [ A(k+1)+ B(k)][(k)(k+1)]
cancel like terms:
1= [ A(k+1)+ B(k)]
solve for k!
To solve for A and B in the equation:
1 A B
---------------- = --- + ---
k(k+1) k k+1
We need to clear the denominators by multiplying both sides of the equation by k(k+1). This gives us:
1 = A(k+1) + B(k)
Next, we can distribute the terms on the right side of the equation:
1 = Ak + A + Bk
Now, we can rearrange the equation to group the variables together:
1 = (A + B)k + A
Since this equation must hold true for all values of k, the coefficients of k and the constant term must be equal. Therefore, we have two equations:
A + B = 0 (coefficients of k)
A = 1 (constant term)
From the first equation, we can solve for A in terms of B:
A = -B
Substituting this into the second equation:
-A = 1
Multiply both sides by -1:
A = -1
Now that we have the value of A, we can substitute it back into the equation A + B = 0:
-1 + B = 0
Solve for B:
B = 1
Therefore, the solutions for A and B are A = -1 and B = 1.