University:
Washington State University
Course:
MATH 172 | Calculus II
Academic year:

2024
Views:

409

Pages:

3
Author:

Bailey Ayala

Question 1 Tell whether the sequence 12, 4, -4, -12, -20, .... is arithmetic. Explain your reasoning. Answer We know that in arithmetic sequence the common difference we can calculate d = a2 - a1 = a3 – a2 =... an - an-1 To check if the given sequence is arithmetic, consider first difference. The following is true d = a2 - a1 = a3 - a2 = a4 - a3 = a5 - a4 = 4 - 12 = -4 - 4 = -12 - (-4) = -20 - (-12) = -8 = -8 = -8 = -8 Since the first difference is constant it implies that the given sequence is arithmetic. Question 2 Flour is sold in 5 kg and 2 kg packets. The 5 kg packets cost $2.75 each, and the 2 kg packets cost $1.25 each. If I boight 65 kg of flour and the total cost was $38.50, how many of each size of packet did I buy? Answer Let x be the number of 5 kg packets of flour and y be the number of 2 kg packets of flour. The total weight bought is 67 kg so we write: 5x + 2y = 67 (1) The total cost is $38.50 so we write: 2.75x + 1.25y = 38.50 (2) Solve the system by elimination. I decided to eliminate y. Multiply (1) by 2.5 to obtain (3): 12.5x + 5y = 167.50 (3) Multiply (2) by 4 to obtain (4): 11x + 5y = 154 (4) Subtract each side of (3) and (4) to eliminate y and solve for x: 1.5x = 13.5 x=9 Solve for yy using (1): 5(9) + 2y = 67 45 + 2y = 67 2y = 22 y = 11 So, you bought 9 packets of 5 kg flour and 11 packets of 2 kg flour. Question 3 At a clothing store, 3 shirts and 8 hats cost $65. The cost for 2 shirts and 2 hats is $30. How much does each shirt and hat cost? Answer Let x be the cost of each shirt and y be the cost of each hat. 3 shirt and 8 hats cost $65: 3x + 8y = 65 2 shirts and 2 hats cost $30: 2x + 2y = 30 Eliminate all possibilities to find a solution. Here, I first eliminated y. Multiply (2) by 4 to obtain (3): 8x + 8y = 120 Subtract (1) and (3) and solve for x: -5x = -55 x = 11 Solve for y using (2): 2(11) + 2y = 30 22 + 2y = 30 2y = 8 y = 4 Hence, each shirt costs $11 and each hat costs $4. Question 4 Jamerra received a $3,000 car loan. She plans on paying off the loan in 2 years. At the end of 2 years, Jamerra will have paid $450 in interest. What is the simple interest rate on the car loan? Answer Use the simple interest formula: I = Prt where I is the interest, P is the principal, r is then rate in decimal, and t is the time in years. Substitute $450, P = $3,000, and t = 2 450 = 3000(r)(2) 450 = 6000r 450/6000 = r r = 0.075 = 7.5% Question 5 Use what you know about bonded, monotonic sequences to show that the following sequences converge an= 3/2(1- 1/3^n ) Answer For the given sequence {an} an+1 - an = 3/2 (1- 1/3^(n+1)) - 3/2 (1- 1/3^n ) = 3/2 ( 1 - 1/3^(n+1) - 1 + 1/3^n) = 3/2 (1/3^n - 1/3^(n+1)) = 3/2 (1/3^n) (1 - 1/3) = 3/2 (1/3^n) (2/3) = 1/3^n >0 Since an+1 > an, therefore, the sequence {an} is monotonic increasing. For all n ≥ 1 (1- 1/3^n) < 1 1/3 * (1 - 1/3^n) < 1/3 an < 1/3 So, the sequence is bounded above with the upper bound 1/3 Thus, being a monotonic increasing sequence bounded above, 1/3 being an upper bound, the given sequence {an} is convergent It completes the proof.