How do you calculate the nth term for 2, 8, 18, 32?

The nth term for the sequence 2, 8, 18, 32 is given by the formula \( n^2 + n \).

To find the nth term of a sequence, we first need to identify a pattern or rule that the terms follow. For the sequence 2, 8, 18, 32, we can start by examining the differences between consecutive terms. The differences are 6, 10, and 14. These differences themselves increase by 4 each time, indicating that the sequence is quadratic.

A quadratic sequence can be expressed in the form \( an^2 + bn + c \). To determine the coefficients \( a \), \( b \), and \( c \), we can set up equations using the first few terms of the sequence. For the first term (n=1), we have:
\[ a(1)^2 + b(1) + c = 2 \]
For the second term (n=2), we have:
\[ a(2)^2 + b(2) + c = 8 \]
For the third term (n=3), we have:
\[ a(3)^2 + b(3) + c = 18 \]

This gives us the system of equations:
\[ a + b + c = 2 \]
\[ 4a + 2b + c = 8 \]
\[ 9a + 3b + c = 18 \]

By solving this system, we find that \( a = 1 \), \( b = 1 \), and \( c = 0 \). Therefore, the nth term formula is \( n^2 + n \). This formula allows us to calculate any term in the sequence by substituting the value of \( n \). For example, for the 4th term (n=4), we get \( 4^2 + 4 = 16 + 4 = 20 \).