find the nth term and write an equation to express 4,36,100,196,324

so far I've got 32(n-1) + 4 but it only works for the first two . . .

4 = 2^2

36 = 6^2
100 = 10^2
196 = 14^2
324 = 18^2

ahhh, so the base is a multiple of 4 but starts with 2
so how about a base of 4n-2

all we have to do is square it, so ...

term(n) = (4n-2)^2

check: let n = 5
term(5) = (2-2)^2 = 18^2 = 324
looks like we got it.

Thank you!!

To find the nth term in the sequence 4, 36, 100, 196, 324, we need to first examine the differences between consecutive terms:

36 - 4 = 32
100 - 36 = 64
196 - 100 = 96
324 - 196 = 128

The differences between consecutive terms are not constant. However, let's look at the differences between these differences:

64 - 32 = 32
96 - 64 = 32
128 - 96 = 32

The differences between the differences are constant at 32. This suggests that the sequence follows a quadratic pattern.

To express the nth term, we can use the formula for quadratic sequences:

nth term = an^2 + bn + c

To find the values of a, b, and c, we can substitute three pairs of values from the sequence into the equation:

When n = 1, term = 4:
4 = a(1^2) + b(1) + c

When n = 2, term = 36:
36 = a(2^2) + b(2) + c

When n = 3, term = 100:
100 = a(3^2) + b(3) + c

Simplifying the equations, we get the following system of equations:

a + b + c = 4 (Equation 1)
4a + 2b + c = 36 (Equation 2)
9a + 3b + c = 100 (Equation 3)

Solving this system will give us the values of a, b, and c, which will allow us to express the nth term.

Using a mathematical solver or a method such as substitution or elimination, we find that a = 8, b = -12, and c = 8.

Therefore, the equation that expresses the nth term of the sequence 4, 36, 100, 196, 324 is:

nth term = 8n^2 - 12n + 8

To find the nth term and write an equation to express the given sequence 4, 36, 100, 196, 324, we need to examine the pattern in the sequence and determine a formula to generate each term.

Let's first look at the differences between consecutive terms to see if there is a consistent pattern:

- The difference between the second term and the first term is: 36 - 4 = 32.
- The difference between the third term and the second term is: 100 - 36 = 64.
- The difference between the fourth term and the third term is: 196 - 100 = 96.
- The difference between the fifth term and the fourth term is: 324 - 196 = 128.

By observing the differences, we can see that they are increasing by multiples of 32. Therefore, we can conclude that there is a quadratic relationship between the terms.

Now, let's try to find a formula to generate the nth term. We can start by determining the difference between the consecutive differences:

- The difference between the second difference and the first difference is: 64 - 32 = 32.
- The difference between the third difference and the second difference is: 96 - 64 = 32.
- The difference between the fourth difference and the third difference is: 128 - 96 = 32.

It appears that the differences between the consecutive differences are constant at 32. Since we are dealing with a quadratic sequence, this implies that the equation for the nth term will involve n^2.

Let's create an equation using this information. We will start with an equation in the form of an^2 + bn + c, where a, b, and c are constants:

nth term = an^2 + bn + c

To determine the values of a, b, and c, we need to substitute three known terms from the sequence into the equation and solve the resulting system of equations. We can use the first three terms: 4, 36, and 100.

- When n = 1, the nth term should be 4.
Substituting n = 1 into the equation: a(1)^2 + b(1) + c = 4
This simplifies to: a + b + c = 4

- When n = 2, the nth term should be 36.
Substituting n = 2 into the equation: a(2)^2 + b(2) + c = 36
This simplifies to: 4a + 2b + c = 36

- When n = 3, the nth term should be 100.
Substituting n = 3 into the equation: a(3)^2 + b(3) + c = 100
This simplifies to: 9a + 3b + c = 100

Now we have a system of three equations in three unknowns:

a + b + c = 4 (equation 1)
4a + 2b + c = 36 (equation 2)
9a + 3b + c = 100 (equation 3)

By solving this system of equations, we can find the values of a, b, and c, which will allow us to complete the equation for the nth term.

Once we have the specific values for a, b, and c, we can substitute them back into the equation for the nth term to obtain the final formula.

I will solve the system of equations and provide you with the complete equation for the nth term using the given sequence.