Algebra

Now apply the identity

We are asked to determine which number divides . Step 1: Check for divisibility by 49 Let's check the values of modulo 49. For small values of : - For , , and . - For , , and . - For , , and . By inspecting values of , we see that the expression is divisible by 49. Thus, the correct answer is .

We are given that must be divisible by 64 for all . Step 1: Simplify the equation First, rewrite the given expression: Note that , so we want to find the behavior of . Step 2: Calculate Since , we find that for any . Thus, the expression becomes: Step 3: Solve for For : Since , the least negative value for is . Thus, the correct answer is .

We are given that the sequence is defined recursively as: We need to find a general expression for . Step 1: Check the base case For : Step 2: Assume the formula holds for Assume that holds for some . Step 3: Prove for Using the recursive formula: Substitute the assumed formula for : Simplify: Thus, the formula holds, and the general expression for is: Thus, the correct answer is .