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

2024
Views:

6

Pages:

2
Author:

Bailey Ayala

Question 1 What is the next number in this sequence: 1,2,6,24,120 ...? Answer 1 The next number is 840. The nth term in the sequence is the smallest number with 2n divisors. Probably the answer they're looking for, though, is 6!=720. Answer 2 After some testing I discovered that these numbers are being multiplied by their corresponding number in the sequence. For example: 1×2=2 2×3=6 6 x 4 = 24 24 x 5 = 120 Which would mean the next number in the sequence would be 120 x 6 = 720 and so on and so forth. For Example: 2!=2×1=2 3! = 3 x 2×1=6 4!=4x3 × 2 × 1 = 24 5!=5×4×3×2×1= 120 6!=6×5×4×3× 2 × 1 = 720 Question 2 What is the least common multiple of 8 and 12? Answer The least common multiple of these two is 24. We should multiple 12 by increasing numbers until we find a number that is also divisible by 8: 12 x 2 = 24, which is also divsible by 8. Question 3 Is this the recursive formula for geometric sequences? {a₁ = first term an=rxan-1 Answer We have {a₁ = first term an = r × an-1 The second term of this sequence is a2 = r × a2-1 ⇒ a2 = r × a1 (2) The third term: a3 = r × a3-1 ⇒ a3 = r × a2 Put value of a2 from equation 2 ⇒ a3 = r × (r× a1) ⇒ a3 = r² × a1 (3) Similarly, you can calculate a4 ⇒ a4 = r × a4-1 ⇒ a4 = r × a3 Put a value of a3 from the equation (3) ⇒ a4 = r × (r² × a1) ⇒ a4 = r³ × a1 (4) And so on. Thus, we have the following a1, r × a1, r² × a1, r³ × a1, .... So we have observed that a2/a1 = a3/a2 = a4/a3=... = an/an-1 = r Thus, it's the formula of geometric sequences. Question 4 What is important about monotonic and bounded sequences? Answer A sequence is said to be monotonic, if it is either monotonically increasing or decreasing. A sequence, an is monotonic increasing if an + 1 ≥ an ∀n ∈ Ν. A sequence, an is monotonic decreasing if an + 1 ≤ an ∀n ∈ N A Sequence, an is bounded if there exist number M>0 such that, an ≤ M ∀n ∈ N By Monotone Convergence Theorem, a monotone sequence of real numbers is convergent if it is bounded. By using this, one can say that every monotone bounded sequence are convergent.