Let α,β,γ and δ be the coefficients of x7,x5,x3 and x respectively in the expansion of (x+x3−1)5+(x−x3−1)5,x>1. If u and v satisfy the equations αu+βvγu+δv=18=20 then u+v equals:
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Step-by-Step Solution
Step 1: Simplify the given expression
We use the binomial expansion formula for (A+B)n+(A−B)n. For n=5, the terms with odd powers of B cancel out. Here, A=x and B=x3−1.
Step 2: Substitute A and B into the simplified expression
Substitute A=x and B=x3−1 into the simplified expression. Then, expand and combine like terms to get the polynomial in descending powers of x.
Step 3: Identify the coefficients
From the expanded polynomial, we identify the coefficients of x7,x5,x3, and x. The coefficient of x7 is α, x5 is β, x3 is γ, and x is δ.
Step 4: Substitute coefficients into the given equations
Substitute the values of α,β,γ, and δ into the given system of linear equations.
Step 5: Solve the system of equations for u and v
Divide equation (1) by 2 to simplify it to 5u+v=9. Divide equation (2) by 10 to simplify it to −2u+v=2. Subtract the second simplified equation from the first to find u. Then substitute the value of u back into one of the simplified equations to find v.
Step 6: Calculate u+v
Finally, add the calculated values of u and v to find their sum.