Algebraic Expressions

Step 1: Express the function in terms of and
We are given the equation: and we need to maximize the function: Using , we rewrite:
Step 2: Differentiate and find the critical points
To maximize , we take the derivative: Setting this to zero: Ignoring (as ), we solve: Thus, and .
Step 3: Compute the Maximum Value
Substituting and into :
Step 4: Compute

Final Answer: .

Step 1: Factor the left-hand side. Step 2: Let’s denote: Observe that all terms in the sum involve powers of 10 decreasing, which suggests this is a number of the form: S = N (a large decimal number)
Step 3: Match the structure of both sides. We are given: So, But rather than compute exactly, we note that the expression on the right is divisible by , and the pattern of the digits in is such that it replicates the decimal expansion of , expanded in base 10 with weighted digits. So the constant acts as the multiplier that "compresses" the expanded decimal form back into powers of 11.
Step 4: Use trial to check for . Try and compute the RHS: Hence, verified that satisfies the identity.