Help please, I don't understand how to do this...
Give an example of an arithmetic sequence that is found in the real world. Find the common difference and write a recursive and iterative rule for the sequence. Use one of the rules to find another term of the sequence.
Then give an example of a geometric sequence that is found in the real world. Find the common ratio and write a recursive and iterative rule for the sequence. Use a rule to find any term.
Then clearly explain why each example is either an arithmetic or geometric sequence.
was up
Sure, I'd be happy to help you understand how to solve this problem!
To start, let's first define what an arithmetic sequence and a geometric sequence are:
- An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.
- A geometric sequence is a sequence of numbers in which each term is found by multiplying the previous term by a constant factor. This constant factor is called the common ratio.
Now let's look for examples of arithmetic and geometric sequences in the real world.
Example 1: Arithmetic Sequence
Suppose you have a runner who runs 5 miles on the first day, and then increases the distance by 2 miles each subsequent day. The distances covered form an arithmetic sequence.
Common Difference: In this case, the common difference is 2, because each term increases by 2 miles.
Recursive Rule: The recursive rule for an arithmetic sequence is a formula that relates each term to the previous term. For this example, the recursive rule would be: a(n) = a(n-1) + 2, where a(n) represents the nth term of the sequence and a(n-1) represents the previous term.
Iterative Rule: The iterative rule for an arithmetic sequence is a formula that allows you to find any term of the sequence without knowing the previous terms. For this example, the iterative rule would be: a(n) = a(1) + (n-1) * 2, where a(n) represents the nth term of the sequence and a(1) represents the first term.
To find another term: Let's say you want to find the distance covered on the 10th day. Using the iterative rule, plug in n=10 and substitute the values of a(1) and the common difference. So, a(10) = 5 + (10-1) * 2 = 5 + 9 * 2 = 23 miles. Therefore, on the 10th day, the runner will cover a distance of 23 miles.
Example 2: Geometric Sequence
Consider a population of bacteria that doubles every hour. The number of bacteria forms a geometric sequence.
Common Ratio: In this case, the common ratio is 2, because each term is obtained by multiplying the previous term by 2.
Recursive Rule: The recursive rule for a geometric sequence is a formula that relates each term to the previous term. For this example, the recursive rule would be: a(n) = a(n-1) * 2, where a(n) represents the nth term of the sequence and a(n-1) represents the previous term.
Iterative Rule: The iterative rule for a geometric sequence is a formula that allows you to find any term of the sequence without knowing the previous terms. For this example, the iterative rule would be: a(n) = a(1) * (2^(n-1)), where a(n) represents the nth term and a(1) represents the first term.
To find any term: Let's say you want to find the number of bacteria after 5 hours. Using the iterative rule, plug in n=5 and the value of a(1). So, a(5) = a(1) * (2^(5-1)) = a(1) * 2^4 = a(1) * 16. Note that in this case, you need to know the value of a(1) to find a(5), which indicates a geometric sequence.
Explanation:
In the arithmetic sequence example, the terms increase by a fixed amount each time, making it an arithmetic sequence.
In the geometric sequence example, the terms are obtained by multiplying the previous term by a fixed ratio, making it a geometric sequence.
By understanding the definitions and formulas for arithmetic and geometric sequences, you can recognize and analyze real-world examples to determine the type of sequence and calculate various terms within the sequence.
thank you so much!! :)
You are welcome :)
simple yearly interest rate i
(percent = 100 i)
at n = 1 first year beginning
Amount = A
second year
Amount = A + Ai
Third year
Amount = A + 2 A i
This is an arithmetic sequence with
A = starting amount
A i = d
so Amount = A + d (n-1)
recursive Amount n = Amount (n-1) + d
where d = Ai
-----------------------------
Geometric sequence = compound interest
Amount at n = (amount at n-1) * (1+i)
so
Amount at n = original amount * (1+i)^(n-1)
where n is 1 at the time of deposit
so if original amount at n = 1 is A
amount
that is of form A r^(n-1)
where r = (1+i)
and n is the number of years starting with deposit at n = 1