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
You're on a roll — here's the solution for the summation: Step 1: Identify the expression and the range of the index r. The summation is given by _r=-1^8 (2r+3). The index r starts at -1 and goes up to 8. Step 2: Calculate the first term (a_1) and the last term (a_n) of the series. For the first term, substitute r=-1: a_1 = 2(-1) + 3 = -2 + 3 = 1 For the last term, substitute r=8: a_n = 2(8) + 3 = 16 + 3 = 19 Step 3: Determine the number of terms (n) in the series. The number of terms is calculated as (upper limit - lower limit + 1): n = 8 - (-1) + 1 = 8 + 1 + 1 = 10 There are 10 terms in this series. Step 4: Use the formula for the sum of an arithmetic series. Since the expression 2r+3 is linear in r, this is an arithmetic series. The sum S_n of an arithmetic series is given by: S_n = (n)/(2)(a_1 + a_n) Substitute the values of n, a_1, and a_n: S_10 = (10)/(2)(1 + 19) S_10 = 5(20) S_10 = 100 The sum is 100. Got more? Send 'em!