write an explicit formula for each sequence. 1,4,7,10
The explicit formula for the sequence 1, 4, 7, 10 is:
an = 3n - 2
where n is the position of the term in the sequence.
For example,
a1 = 3(1) - 2 = 1
a2 = 3(2) - 2 = 4
a3 = 3(3) - 2 = 7
a4 = 3(4) - 2 = 10
and so on.
is that correct
Yes, that is correct.
show your work
To find the explicit formula for the sequence 1, 4, 7, 10, we can first observe that the difference between each consecutive term is 3.
So, we can write the difference as:
d = 4 - 1 = 7 - 4 = 10 - 7 = 3
Since the common difference is constant, we know that we are dealing with an arithmetic sequence, and we can use the formula for the nth term of an arithmetic sequence:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, n is the position of the term in the sequence, and d is the common difference.
Now, we can substitute the given values:
a1 = 1
d = 3
So,
an = 1 + (n-1)(3)
= 1 + 3n - 3
= 3n - 2
Therefore, the explicit formula for the sequence 1, 4, 7, 10 is:
an = 3n - 2.
To find an explicit formula for the sequence 1, 4, 7, 10, we need to observe the pattern and determine a formula that will generate each term. In this case, it is an arithmetic sequence, as the difference between consecutive terms is constant (3).
Let's call the first term of the sequence "a" and the common difference "d". The formula for an arithmetic sequence is:
An = a + (n - 1) * d
In our case, a = 1 and d = 3. Plugging these values into the formula, we get:
An = 1 + (n - 1) * 3
Simplifying further:
An = 1 + 3n - 3
Finally, we can simplify this expression:
An = 3n - 2
Therefore, the explicit formula for the sequence 1, 4, 7, 10 is An = 3n - 2.
To find an explicit formula for the given sequence, we need to determine the pattern and use it to define a formula. Here are the steps to find an explicit formula for the sequence 1, 4, 7, 10:
1. Identify the difference between consecutive terms:
The difference between the second term (4) and the first term (1) is 3.
The difference between the third term (7) and the second term (4) is also 3.
Similarly, the difference between the fourth term (10) and the third term (7) is 3.
2. Verify if the differences are consistent:
Since the differences are the same for all consecutive terms, we can conclude that this is an arithmetic sequence.
3. Determine the common difference (d):
As mentioned earlier, the common difference between consecutive terms is 3.
4. Write the explicit formula for an arithmetic sequence:
The explicit formula for an arithmetic sequence is given by the equation:
an = a1 + (n - 1) * d
where:
- an represents the nth term of the sequence,
- a1 is the first term of the sequence,
- n is the position of the term,
- d is the common difference between consecutive terms.
5. Substitute the known values into the explicit formula:
In this case, a1 = 1 (first term), d = 3 (common difference), n represents the position of the term.
Applying the values in the formula, we get:
an = 1 + (n - 1) * 3
Therefore, the explicit formula for the given sequence 1, 4, 7, 10 is:
an = 1 + 3(n - 1)