need help asap!

Question

1. Repeating decimals.

1. a. Is 0.9 equal to 1? Explain your reasoning using complete sentences.
2. b. List another repeating decimal that is equal to an integer.

2. Comparing explicit and recursive rules.

1. a. Write an explicit rule and recursive rule for the sequence of numbers 3, 8, 13, 18, 23,…
2. b. Find the 1,000th term of the sequence.
3. c. Which rule did you use to find the 1,000th term? Why did you choose that rule?
4. d. What is a disadvantage of a recursive rule when compared to an explicit rule?

Answer 1

N = 0.9999999999......

10N = 9.999999999........

10N - N = 9

9N = 9

N = 1

Note: this is true only if the 9's never stop, which makes it slilghtly impossible to check!

Answer 2

https://opt-storage-prod.s3.amazonaws.com/shrine/store/6f3e55b3411e143ce8843b2342350340db369e424703db76718f925e49ac9cbb.JPG

Follow the link above for a very helpful graphic of explicit and recursive rule forms for arithmetic and geometric sequences.

1. Our first step is to determine whether this is an arithmetic or geometric sequence
2. 3, 8, 13, 18, 23
3. The obvious pattern is that we add 5 to get from number to number therefore it is arithmetic (geometric involves multiplication)
4. Explicit and recursive forms are a bit confusing to look at but it's really just a matter of plugging in the right numbers into the right form.
5. Variables
6. a_n = the nth number in the sequence. e.g. the fifth term would be a₅
7. a_{1 = the first term in the sequence}
8. In this that would be 3
9. a_{n-1 =} reverse to the term one before the term we are asked to evaluate
10. For a₅, a_n-1 would therefore be a₄
11. d = the common difference (what we are adding or multiplying)
12. In this case that would be 5
13. n = nth term we are asked to solve for
14. In this case that will be 1000

1. Explicit
2. a_{n =} a₁ + d(n-1)
3. Plug in a₁ and d
4. Recursive
5. Two parts written in brackets
6. a₁ = first term
7. a_{n =} a_n-1 + d(n-1)
8. plug in plug in our a₁ and d, we plug in a number for a_n-1 if we are asked to apply the rule to a particular term
9. Now plug in 1000 for n and solve
10. Because the recursive rule is applied in reference to the previous term (a_n-1) the explicit rule is easier to apply. We can find the 1000th term without knowing the 999th term. For higher ns the explicit rule is better.