The number of integral terms in the expansion of (521+781)1016 is:
Get the complete, step-by-step math solution for: "The number of integral terms in the expansion of (1)/(5²) + (1)/(7^8) ^{1016} is:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Write the general term of the binomial expansion
We start by writing the general term of the binomial expansion (a+b)n. In this problem, a=521, b=781, and n=1016. The general term Tr+1 will help us identify which terms are integral.
Step 2: Substitute values into the general term
Now, we substitute the given values of a, b, and n into the general term formula. This gives us the specific form of each term in the expansion.
Step 3: Simplify the general term
We simplify the expression by using the property (xm)n=xmn and 1/xk=x−k. For a term to be integral, the powers of 5 and 7 must be non-negative integers. Since the powers are 2r−2032 and −8r, for the term to be integral, both 2r−2032≥0 and −8r≥0 must hold.
Step 4: Determine the range of 'r' for integral terms
From the conditions derived in the previous step, we solve for r. The first inequality 2r−2032≥0 implies r≥1016. The second inequality −8r≥0 implies r≤0.
Step 5: Find common values of 'r'
We need to find values of r that satisfy both r≥1016 and r≤0. Since r must also be an integer between 0 and n (inclusive), i.e., 0≤r≤1016, there is no integer value of r that can satisfy both conditions simultaneously.