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

2024
Views:

45

Pages:

2
Author:

Bailey Ayala

Question 1 (a) What are the conditions on the common ratio r and initial value a, that would make the resulting geometric sequence increasing? (b) What are the conditions on the common ratio r and initial value a, that would make the resulting geometric sequence decreasing? (c) What are the conditions on the common difference d and initial value a, that would make the resulting arithmetic sequence increasing? Answer a) It is required to explain the condition on common ratio r and initial value a to make the resulting geometric sequence increasing. General geometric sequence with common ratio r and initial value a is represented as: a, ar, ar², ar³, .....arⁿ For this sequence to be increasing: a < ar < ar² < ar³ <... < arⁿ⁻¹ < arⁿ Now a < ar a/a < r ⇒ 1 < r Thus, for geometric sequence to be increasing the condition on common ratio r should be greater than or equals to 1, and no condition om initial value a is required. b) It is required to explain the condition on common ratio r and initial value a to make the resulting geometric sequence is decreasing. a, ar, ar², ar³,... arⁿ to be decreasing: a > ar > ar² ≥ ar³ > ... > arⁿ⁻¹ > arⁿ a > ar a/a > r => 1 > r Thus, for geometric sequence to be decreasing the condition on common ratio r that it should be less than or equals to 1, and no condition on initial value a is required. c) It is required to explain the condition on the common difference d and initial value a, that would make the resulting arithmetic sequence increasing Note that a general arithmetic sequence with common difference d and initial value a is represented as: a, a + d, a + 2d, a + 3d,..., a+nd To be increasing: a ≤ a + d ≤ a + 2d ≤ a + 3d < ... < a + (n - 1)d ≤ a + nd a ≤ a + d ⇒ a-a< d ⇒ 0 < d Thus, for arithmetic sequence to be increasing the condition on common difference d that it should be greater than or equals to 0, and no condition on initial value a is required. Question 2 The recursive formula A(n) = A(n - 1) + 5 represents the amount of money in your savings account after n weeks. You start with $100. What is an explicit formula for the amount of money in your savings account after n weeks? A. A(n) = 5 + 100(n - 1) B. A(n) = 100(n - 1) - 5 C. A(n) = 100 + 5(n - 1) D. A(n) = 5(n - 1) – 100 Answer The recursive formula is for arithmetic sequence where A1 is the first term (A1 = 100) and the common difference is d = 5. The explicit formulf is given by: A(n) = A1 + (n-1)d So, we have: A(n) = 100 + (n - 1)(5) A(n) = 100 + 5(n - 1) So, the correct answer is choice C. Question 3 Write a formula for the nth term of the arithmetic sequence 15, 20, 25, 30, ... Answer The nnth term of an arithmetic sequence is an = a₁ + (n - 1)d where a₁ is the first term and d is the common difference. For the arithmetic sequence 15, 20, 25, 30, ... the first term is a₁ = 15 and the common difference is d = a2 - a1 = 20 - 15 = 5. The nth term is then: an = a1 + (n-1)d an = 15 + (n - 1)(5) Substitute. an = 15 + 5n - 5 Distribute. an = 10 + 5n Combine like terms.