a student write that an=3n+1 is a explicit formula for the sequence 1,4,7,10.explain the error and write the correct explicit formula for the sequence.

if the terms are numbered from 1, then

an = 3(n-1)+1 = 3n-2

The student has made an error in this case.

The given sequence is 1, 4, 7, 10. If we try to apply the formula an = 3n + 1 to these numbers, we find that it does not generate the correct terms. Let's check:

For n = 1, using an = 3n + 1, we get a1 = 3(1) + 1 = 4, which is correct.
For n = 2, using an = 3n + 1, we get a2 = 3(2) + 1 = 7, which is correct.
For n = 3, using an = 3n + 1, we get a3 = 3(3) + 1 = 10, which is correct.

However, the formula fails to produce the correct terms for all subsequent n values. For example, using n = 4, we should get a4 = 3(4) + 1 = 13, but the given sequence has 10 at that position.

To determine the correct explicit formula for the sequence, we can observe that each term increases by a constant difference of 3. Hence, the formula for this sequence should be:

an = 3n - 2

By substituting the values of n into this formula, we can verify that it generates the correct terms:

For n = 1, we get a1 = 3(1) - 2 = 1, which is correct.
For n = 2, we get a2 = 3(2) - 2 = 4, which is correct.
For n = 3, we get a3 = 3(3) - 2 = 7, which is correct.
For n = 4, we get a4 = 3(4) - 2 = 10, which is correct.

Therefore, the correct explicit formula for the given sequence is an = 3n - 2.

The student made an error in claiming that the formula "an = 3n + 1" is an explicit formula for the given sequence 1, 4, 7, 10.

To determine if the formula is correct, we can test it by substituting the values of n from the given sequence and checking if we obtain the corresponding terms.

Using the formula an = 3n + 1, let's substitute the values of n from the given sequence:
When n = 1: a1 = 3(1) + 1 = 4
When n = 2: a2 = 3(2) + 1 = 7
When n = 3: a3 = 3(3) + 1 = 10

As you can see, the formula "an = 3n + 1" does indeed produce the correct values for the given sequence 1, 4, 7, 10. Therefore, the student's claim is correct, and the explicit formula for the sequence is indeed "an = 3n + 1."