The simple formula for the nth term of an arithmetic sequence is an = 4n + 16. What is the explicit formula corresponding to the simple formula?

The simple formula for the nth term of an arithmetic sequence is an=5n+ 15. What is the explicit formula corresponding to the simple formula?

To find the explicit formula for an arithmetic sequence, we need to express the nth term (an) in terms of the first term (a₁) and the common difference (d).

In this case, the given simple formula is an = 4n + 16.

Comparing this to the general form of the explicit formula, which is an = a₁ + (n-1)d, we can determine the values of a₁ and d.

From the given formula, we see that the first term (a₁) is obtained when n = 1. Plugging in n = 1, we get:

a₁ = 4(1) + 16
a₁ = 4 + 16
a₁ = 20

Thus, the value of the first term (a₁) is 20.

Next, we can find the common difference (d) by comparing terms. Subtracting the (n-1)th term from the nth term, we have:

an - a(n-1) = (4n + 16) - (4(n-1) + 16)
= (4n + 16) - (4n - 4 + 16)
= (4n + 16) - (4n + 12)
= 4 (constant difference)

Hence, the common difference (d) is 4.

Now that we have determined the values of a₁ and d, we can plug them back into the general explicit formula to find the specific explicit formula for the given arithmetic sequence.

Therefore, the explicit formula is: an = 20 + (n-1)4.