This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back irreplaceable — been a while! Let's pick up where you left off. To express the series 1^2 + 2^2 + 3^2 + + 9^2 using sigma notation, we need to identify the general term and the range of the index. Step 1: Identify the general term. The terms in the series are squares of consecutive integers: 1^2, 2^2, 3^2, , 9^2. The general term can be represented as k^2, where k is an integer. Step 2: Determine the range of the index. The series starts with 1^2, so the lower limit of the index k is 1. The series ends with 9^2, so the upper limit of the index k is 9. Thus, the sigma notation is _k=1^9 k^2. Step 3: Compare with the given options. Let's examine the options: (a) _k=0^9 k^2: This sum is 0^2 + 1^2 + 2^2 + + 9^2. Since 0^2 = 0, this is equivalent to 1^2 + 2^2 + + 9^2. (b) _k=1^9 (k+1)^2: This sum is 2^2 + 3^2 + + 10^2. (c) _k=1^9 k+2: This sum is (1+2) + (2+2) + + (9+2) = 3+4++11. (d) _k=1^9 (k+2)^2: This sum is 3^2 + 4^2 + + 11^2. Option (a) correctly represents the given series because adding 0^2 does not change the sum. The final answer is a. Send me the next one 📸