thank you Steve for helping me out.
Accurately classify the sequence and then create a corresponding explicit and recursive formula for finding the nth terms of each sequence.
A. 32, 30, 28, 26,.....
B. 9,-3, 1, - _1 .......
surely you can do A. A simple arithmetic progression.
I suspect B is supposed to be
9,-3,1,-1/3,...
a simple geometric sequence.
You have handy formulas for these things.
To classify the sequence and find the explicit and recursive formulas for each, we need to look for patterns or relationships between the terms in each sequence.
Let's start with sequence A: 32, 30, 28, 26, ...
Sequence A seems to be decreasing by 2 with each term. So the explicit formula for this sequence can be written as:
An = 32 - 2n
where n is the position of the term in the sequence. For example, if we want to find the 5th term, we substitute n = 5 into the formula:
A5 = 32 - 2(5) = 32 - 10 = 22
Therefore, the 5th term of sequence A is 22.
Now let's move on to sequence B: 9, -3, 1, -1, ...
In this sequence, the terms seem to alternate between positive and negative numbers. It's also decreasing by 4 with each term. To represent this pattern, we can use a recursive formula.
Let's define Bn as the nth term of sequence B. By observing the pattern, we can see that when n is even, the term is positive, and when n is odd, the term is negative. Therefore, we can define the recursive formula for sequence B as follows:
B1 = 9 (the first term is 9)
Bn = Bn-1 - 4(-1)^(n-1)
This formula states that each term (starting from the second term) is equal to the previous term minus 4 times (-1) raised to the power of (n-1).
For example, to find the 4th term (B4) in sequence B, we can use the recursive formula:
B4 = B3 - 4(-1)^(4-1)
= B3 - 4(-1)^3
= B3 + 4
= 1 + 4
= 5
Therefore, the 4th term of sequence B is 5.
By using these formulas, you can find the nth term of each sequence without having to compute all the preceding terms.