1.write the first four terms of the sequence that begins with 2000 and has the common ratio 1.05.
2.write the first four terms of the sequence that begins with 5000 and decays 15% with each term.what is the common ratio?
3.write a recursive formula for the sequence that begins 1250,1350,1458,1574.64,....
1. trivial - what did you get?
2. ratio is 0.85
3. since r = 1.08, what do you get?
1. To find the terms of a sequence that begins with 2000 and has a common ratio of 1.05, we can use the formula:
Term(n) = Starting Term * (Common Ratio)^(n-1)
Therefore, the first four terms are:
Term(1) = 2000 * (1.05)^(1-1) = 2000
Term(2) = 2000 * (1.05)^(2-1) = 2100
Term(3) = 2000 * (1.05)^(3-1) = 2205
Term(4) = 2000 * (1.05)^(4-1) = 2315.25
2. To find the terms of a sequence that begins with 5000 and decays 15% with each term, we can use the formula:
Term(n) = Starting Term * (1 - Decay Rate)^(n-1)
Here, the decay rate is 15% or 0.15.
Therefore, the first four terms are:
Term(1) = 5000 * (1 - 0.15)^(1-1) = 5000
Term(2) = 5000 * (1 - 0.15)^(2-1) = 4250
Term(3) = 5000 * (1 - 0.15)^(3-1) = 3612.50
Term(4) = 5000 * (1 - 0.15)^(4-1) = 3070.63
The common ratio in this case is (1 - Decay Rate), which is 0.85.
3. To write a recursive formula for the sequence that begins with 1250, 1350, 1458, 1574.64, ..., we observe that each term is obtained by multiplying the previous term by a constant factor (common ratio).
Let's define the recursive formula as Term(n) = Term(n-1) * r, where r is the common ratio.
Therefore, the recursive formula for this sequence would be:
Term(1) = 1250 (given)
Term(n) = Term(n-1) * r
The common ratio, r, can be found by dividing any term by its previous term. In this case, let's find it for the second and third terms:
1350 / 1250 = 1.08
1458 / 1350 = 1.08
So, the common ratio, r, is 1.08.
Therefore, the recursive formula for the sequence is:
Term(1) = 1250 (given)
Term(n) = Term(n-1) * 1.08
1. To find the first four terms of a sequence that begins with 2000 and has a common ratio of 1.05, we can use the formula for a geometric sequence: an = a1 * r^(n-1), where "an" represents the nth term, "a1" is the first term, "r" is the common ratio, and "n" is the term number.
In this case, the first term (a1) is 2000 and the common ratio (r) is 1.05. Let's substitute these values into the formula to find the first four terms:
a1 = 2000
r = 1.05
a2 = 2000 * (1.05)^(2-1) = 2000 * 1.05 = 2100
a3 = 2000 * (1.05)^(3-1) = 2000 * 1.05^2 = 2205
a4 = 2000 * (1.05)^(4-1) = 2000 * 1.05^3 = 2315.25
Therefore, the first four terms of the sequence are 2000, 2100, 2205, and 2315.25.
2. To find the first four terms of a sequence that begins with 5000 and decays 15% with each term, we can also use the formula for a geometric sequence. However, we need to find the common ratio first.
The formula for a decaying sequence is: an = a1 * (1 - d)^n-1, where "an" represents the nth term, "a1" is the first term, "d" is the decay rate (expressed as a decimal), and "n" is the term number.
In this case, the first term (a1) is 5000, and the decay rate (d) is 15%, which is equivalent to 0.15. Let's substitute these values into the formula to find the first four terms:
a1 = 5000
d = 0.15
a2 = 5000 * (1 - 0.15)^(2-1) = 5000 * 0.85 = 4250
a3 = 5000 * (1 - 0.15)^(3-1) = 5000 * 0.7225 = 3612.5
a4 = 5000 * (1 - 0.15)^(4-1) = 5000 * 0.614125 = 3070.625
Therefore, the first four terms of the sequence are 5000, 4250, 3612.5, and 3070.625. The common ratio in this case is (1 - d), which equals 0.85.
3. To write a recursive formula for the sequence that begins with 1250, 1350, 1458, 1574.64, and so on, we need to identify the pattern between consecutive terms.
Looking at the terms, we can observe that each term is obtained by multiplying the previous term by the same factor, which is approximately 1.08.
So, the recursive formula for this sequence can be written as:
a1 = 1250
an = an-1 * 1.08
This formula states that the first term (a1) is 1250, and each subsequent term (an) is obtained by multiplying the previous term (an-1) by 1.08.