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.
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You're on a roll — here are the solutions for question 4. 4) Find the sum of the first 10 terms of the following sequences a) 8, 3, -2, -7, ... Step 1: Identify the type of sequence and its parameters. As determined in question 3a, this is an arithmetic sequence with the first term a_1 = 8 and common difference d = -5. We need to find the sum of the first 10 terms, so n=10. Step 2: Use the formula for the sum of the first n terms of an arithmetic sequence. The formula is S_n = (n)/(2)(2a_1 + (n-1)d). Substitute n=10, a_1=8, and d=-5: S_10 = (10)/(2)(2(8) + (10-1)(-5)) S_10 = 5(16 + (9)(-5)) S_10 = 5(16 - 45) S_10 = 5(-29) S_10 = -145 The sum of the first 10 terms is -145. b) 3, 1, (1)/(3), (1)/(9), ... Step 1: Identify the type of sequence and its parameters. As determined in question 3b, this is a geometric sequence with the first term a_1 = 3 and common ratio r = (1)/(3). We need to find the sum of the first 10 terms, so n=10. Step 2: Use the formula for the sum of the first n terms of a geometric sequence. The formula is S_n = (a_1(1-r^n))/(1-r). Substitute n=10, a_1=3, and r=(1)/(3): S_10 = (3(1 - (1)/(3))^10)1 - (1)/(3) Step 3: Simplify the expression. S_10 = (3(1 - 1)/(3^10))(2)/(3) S_10 = 3(3^10-13^10)(2)/(3) S_10 = 3 · 3^10-13^10 · (3)/(2) S_10 = 3^2(3^10-1)2 · 3^10 S_10 = 9(3^10-1)2 · 3^10 S_10 = (9(59049-1))/(2 · 59049) S_10 = (9(59048))/(118098) S_10 = (531432)/(118098) S_10 = (265716)/(59049) This fraction can be simplified by dividing the numerator and denominator by 9: S_10 = (59048)/(13122) Further simplification by dividing by 2: S_10 = (29524)/(6561) The sum of the first 10 terms is (29524)/(6561). What's next?