what is the explicit formula of the sequence 3,1,-1,-3,-5
To find the explicit formula of the sequence 3, 1, -1, -3, -5, we need to look for a pattern in the terms.
The difference between consecutive terms is decreasing by 2. It starts with -2, then becomes -4, -6, and so on. So, the common difference can be represented as d = -2.
The first term of the sequence is 3. To find the explicit formula, we can use the formula for the nth term of an arithmetic sequence:
An = A1 + (n - 1) * d
Substituting the known values, we have:
An = 3 + (n - 1) * (-2)
Simplifying:
An = 3 - 2n + 2
Finally, we can rewrite the formula:
An = -2n + 5
Therefore, the explicit formula for the sequence 3, 1, -1, -3, -5 is An = -2n + 5.
To find the explicit formula of a sequence, we need to look for a pattern in the numbers. This sequence appears to be decreasing by 2 each time. Let's examine the differences between consecutive terms:
1st term to 2nd term: 1 - 3 = -2
2nd term to 3rd term: -1 - 1 = -2
3rd term to 4th term: -3 - (-1) = -2
4th term to 5th term: -5 - (-3) = -2
We notice that the differences between consecutive terms are constant and equal to -2.
Now, let's assume that the nth term of the sequence is denoted by the variable a_n. We can express the pattern we observed using this notation:
a_1 = 3
a_2 = 1 (a_1 - 2)
a_3 = -1 (a_2 - 2)
a_4 = -3 (a_3 - 2)
a_5 = -5 (a_4 - 2)
From here, we can observe that each term is obtained by subtracting 2 from the previous term. Therefore, we can write the explicit formula for this sequence as:
a_n = 3 - 2(n - 1)
In this formula, n represents the position of the term in the sequence.
I think each term is the previous term minus 2. So it's an arithmetic sequence where a=3 and r= -2. So the nth term (starting with n=0) of the sequnce is
3 - 2n