if the sequence is

2,15,52,125,246....
find the nth term expression
and sum of the nth terms

see related questions below.

i reposted these because i don notn know what u mean by

ur very last step 4th polynomial something

take a look at the sums of powers of integers:

1+2+3+... = n(n-1)/2 (2nd degree)
1^2+2^2+3^2+... = n(n+1)(2n+1)/6 (3rd degree)
1^3+2^3+... = n^2(n+1)^2/4 (4th degree)

The sum of kth powers of integers is a k+1 degree polynomial.

To find the nth term expression and sum of the nth terms for the given sequence 2, 15, 52, 125, 246..., we need to analyze the pattern and try to find a general formula.

Let's observe the differences between consecutive terms:
15 - 2 = 13,
52 - 15 = 37,
125 - 52 = 73,
246 - 125 = 121.

We notice that these differences are increasing by 24 each time:
13 + 24 = 37,
37 + 24 = 73,
73 + 24 = 97.

Let's write down the differences in a separate sequence to observe them better: 13, 37, 73, 121, 169...

Now, let's find the differences between these differences:
37 - 13 = 24,
73 - 37 = 36,
121 - 73 = 48,
169 - 121 = 48.

We observe that these differences between differences are constant, which implies that the original sequence has a quadratic nature.

The formula to find the nth term of a quadratic sequence is:
nth term = an^2 + bn + c,

where a, b, and c are constants to be determined.

To find the values of a, b, and c, we can substitute some values of the sequence into the formula and solve a system of equations.

Using the first three terms:
2 = a(1^2) + b(1) + c,
15 = a(2^2) + b(2) + c,
52 = a(3^2) + b(3) + c.

Simplifying these equations, we get:
a + b + c = 2, (Equation 1)
4a + 2b + c = 15, (Equation 2)
9a + 3b + c = 52. (Equation 3)

To solve this system of equations, we can use any method of solving simultaneous equations, such as substitution or elimination.

Let's use the elimination method to find the values of a, b, and c:

Subtracting Equation 1 from Equation 2, we get:
3a + b = 13. (Equation 4)

Subtracting Equation 1 from Equation 3, we get:
8a + 2b = 50. (Equation 5)

Multiplying Equation 4 by 2, we get:
6a + 2b = 26. (Equation 6)

Now, subtracting Equation 5 from Equation 6, we can eliminate the variable b:
6a + 2b - (8a + 2b) = 26 - 50,
-2a = -24,
2a = 24,
a = 12.

Substituting the value of a into Equation 4, we get:
3(12) + b = 13,
36 + b = 13,
b = 13 - 36,
b = -23.

Substituting the values of a and b into Equation 1, we get:
12 + (-23) + c = 2,
-11 + c = 2,
c = 2 + 11,
c = 13.

Therefore, the nth term expression for the given sequence is:
nth term = 12n^2 - 23n + 13.

To find the sum of the nth terms, we need to use the formula for the sum of a quadratic sequence.

The sum of the n terms in a quadratic sequence is given by:
Sum of nth terms = n/6 * (2an^2 + 3bn + 6c).

Using the values of a, b, and c that we found earlier (a = 12, b = -23, c = 13), we can substitute them into the formula.

Hence, the sum of the nth terms is:
Sum of nth terms = n/6 * (24n^2 - 69n + 78).