The nth term of the series
The given series has a sum of n terms, which is represented as 5n^2 + 2n. We need to find the nth term of this series.

Understanding the series
To find the nth term of the series, we need to understand the pattern in the given sum. Let's analyze the sum 5n^2 + 2n:

- The first term (n = 1) is 5(1)^2 + 2(1) = 7.
- The second term (n = 2) is 5(2)^2 + 2(2) = 24.
- The third term (n = 3) is 5(3)^2 + 2(3) = 47.
- The fourth term (n = 4) is 5(4)^2 + 2(4) = 76.
- And so on.

Finding the pattern
By observing the given series, we can notice that the difference between consecutive terms is increasing by 5. Let's calculate the differences:

- The difference between the first and second terms is 24 - 7 = 17.
- The difference between the second and third terms is 47 - 24 = 23.
- The difference between the third and fourth terms is 76 - 47 = 29.

We can see that the differences are increasing by 6 each time. This suggests that the coefficients of n^2 and n in the nth term will also be increasing by a constant value.

Deriving the nth term
Let's assume the nth term of the series is represented as An^2 + Bn + C, where A, B, and C are constants to be determined.

Since the difference between consecutive terms is increasing by 5, we can write the equation:

An^2 + Bn + C - (A(n-1)^2 + B(n-1) + C) = 5

Simplifying the equation, we get:

An^2 + Bn + C - (A(n^2 - 2n + 1) + B(n-1) + C) = 5

An^2 + Bn + C - (An^2 - 2An + A + Bn - B + C) = 5

Cancelling out the common terms, we get:

An^2 + Bn + C - An^2 + 2An - A - Bn + B + C = 5

Simplifying further, we get:

2An + 2An - A + B - Bn + B = 5

4An - Bn + 2An - A + B = 5

(4A - B)n + (2A - A + B) = 5

Comparing the coefficients of n and the constant term on both sides of the equation, we get:

4A - B = 0 ----(1)
2A - A + B = 5 ----(2)

Solving equations (1) and (2), we find:

A = 1/2
B = 2

Therefore, the nth term of the series is:

An^2 + Bn + C = (1/2)n^2 +