Sandra wrote the sequence shown here. Which equation represents the rule for finding the nth term of this sequence?

2, 5, 10, 17, ...

what does 3,5,7 mean?

so the second term is the first term +2+1
the third (n) term is the second term+5

termn= termn-1 + (2n-1)

check: 4th term= third term +
2n-1
17=? 10 +8-1=17 Yea

notice each value in the sequence is 1 larger than the perfect square, so

tn = n^2 + 1

To find the rule for finding the nth term of this sequence, let's analyze the pattern between each term:

The difference between the terms is increasing. We can notice that the difference between the 1st and 2nd term is 5 - 2 = 3, the difference between the 2nd and 3rd term is 10 - 5 = 5, and the difference between the 3rd and 4th term is 17 - 10 = 7.

Since the difference is increasing by 2 each time, we can assume that the sequence follows a quadratic rule. Let's test it.

We can represent the general form of a quadratic sequence as:

nth term = an^2 + bn + c

To find the values of a, b, and c, we can substitute the values from the given sequence:

1st term (n = 1): 2 = a(1)^2 + b(1) + c
2nd term (n = 2): 5 = a(2)^2 + b(2) + c
3rd term (n = 3): 10 = a(3)^2 + b(3) + c

Now we have a system of three equations:

a + b + c = 2 -> Equation 1
4a + 2b + c = 5 -> Equation 2
9a + 3b + c = 10 -> Equation 3

We can solve this system of equations to find the values of a, b, and c. Let me calculate it for you.

Solving these equations, we find:
a = 2/3, b = 2/3, c = 4/3

Now we can substitute these values back into the general quadratic equation:

nth term = (2/3)n^2 + (2/3)n + (4/3)

So, the equation representing the rule for finding the nth term of this sequence is:
nth term = (2/3)n^2 + (2/3)n + (4/3)