How do you find the nth term in a sequence of numbers if the values are increasing then decreasing? Ex. 1: -2, 3: 10, 5: -18

There are probably more than one "correct" answer.

1: -2, 3: 10, 5: -18
I think of it as two separate series interleaved together, namely:
1, 3, 5, 7, 9....
and
-2, 10, -18, 26, -34, ....
Your teacher has the final word.

To find the nth term in a sequence of numbers when the values are increasing then decreasing, you can use a method called interpolation. Interpolation is a technique that allows you to estimate values between known data points.

In this case, we have some data points given to us:
1st term: -2
3rd term: 10
5th term: -18

Step 1: Identify the pattern:
To see the pattern more clearly, let's write the given terms with their corresponding positions as:
1st term (-2)
3rd term (10)
5th term (-18)

It seems that the pattern alternates between increasing and decreasing values. We can also observe that the difference between the terms is not constant. For example, the difference between the 1st term and the 3rd term is 12, while the difference between the 3rd term and the 5th term is -28.

Step 2: Find the common difference between consecutive terms:
To find the common difference between consecutive terms, subtract the previous term from the next term.

Difference between the 1st and 3rd terms = 10 - (-2) = 12
Difference between the 3rd and 5th terms = -18 - 10 = -28

Since the differences are not constant, it indicates that the sequence is not straightforward linear or arithmetic. Therefore, we need to use a more advanced method like interpolation.

Step 3: Use interpolation to estimate the missing terms:
Interpolation involves estimating the missing terms by finding a relationship or equation that connects the given terms. In this case, it appears that the pattern alternates between increasing and decreasing terms.

One possible approach to interpolate the missing terms is to use quadratic interpolation, assuming that the pattern can be represented by a quadratic equation. Quadratic equations have the general form: ax^2 + bx + c, where x represents the position of the term.

We can write the equation for the sequence as:
a(x^2) + bx + c

Since we have three unknowns (a, b, and c), we need three equations to solve for these variables. We can use the data points we already have to create the equations and solve them simultaneously.

Using the given data points:
1. (-2) = a(1^2) + b(1) + c
2. (10) = a(3^2) + b(3) + c
3. (-18) = a(5^2) + b(5) + c

By solving this system of equations simultaneously, we can find the values of a, b, and c.
Once we find a, b, and c, we can substitute the value of n into the quadratic equation to determine the nth term.