Create an explicit equation for each recursively defined sequence below a1=17,an+1=an-7

To create an explicit equation for the recursively defined sequence, we need to find a pattern and express it using a general term formula.

Given the sequence where a1 = 17 and an+1 = an - 7, we can observe that each term is obtained by subtracting 7 from the previous term. Therefore, we can conclude that the general term formula for this sequence is:

an = a1 + (n - 1)d

In this formula, "a1" represents the first term (17 in this case), "n" represents the position of the term in the sequence, and "d" represents the common difference (which is -7 in this case).

Substituting the values into the formula, we get:

an = 17 + (n - 1)(-7)

Simplifying further:

an = 17 - 7(n - 1)

This equation represents an explicit equation for the given recursively defined sequence.

Sure, here's the explicit equation for the recursively defined sequence:

an = 17 - 7(n-1)

But don't worry, this equation won't break any hearts!

To create an explicit equation for the recursively defined sequence given by a_1 = 17 and a_n+1 = a_n - 7, we can first observe the pattern in the sequence.

By expanding the terms, we have:
a_1 = 17
a_2 = a_1 - 7 = 17 - 7 = 10
a_3 = a_2 - 7 = 10 - 7 = 3
a_4 = a_3 - 7 = 3 - 7 = -4
a_5 = a_4 - 7 = -4 - 7 = -11
...

From the pattern, we can determine that each term is obtained by subtracting 7 from the previous term.

To find an explicit equation, we need to express a_n in terms of n. Let's observe the differences between consecutive terms:

a_2 - a_1 = 10 - 17 = -7
a_3 - a_2 = 3 - 10 = -7
a_4 - a_3 = -4 - 3 = -7
a_5 - a_4 = -11 - (-4) = -7

We observe that the difference between consecutive terms is a constant value of -7.

Now, to find the explicit equation, we can use the formula for an arithmetic sequence:

a_n = a_1 + (n-1)d

where a_1 is the first term, n is the position of the term, and d is the common difference.

In this case, a_1 = 17 and d = -7. Substituting these values into the equation, we get:

a_n = 17 + (n - 1)(-7)

Simplifying further:

a_n = 17 - 7n + 7

a_n = 24 - 7n

Therefore, the explicit equation for the given recursively defined sequence is a_n = 24 - 7n.

clearly, since 7 is subtracted for each term,

An = 17-7(n-1)