How do you find the nth term of 2, 10, 26, 50?

To find the nth term of the sequence 2, 10, 26, 50, use the formula \( T_n = 2n^2 + n \).

To understand how to derive this formula, let's first look at the given sequence: 2, 10, 26, 50. We need to determine the pattern or rule that generates these numbers. One way to do this is by examining the differences between consecutive terms.

First, calculate the first differences:
10 - 2 = 8,
26 - 10 = 16,
50 - 26 = 24.

Next, calculate the second differences:
16 - 8 = 8,
24 - 16 = 8.

Since the second differences are constant (8), this indicates that the sequence is quadratic. A quadratic sequence can be expressed in the form \( T_n = an^2 + bn + c \).

To find the coefficients \( a \), \( b \), and \( c \), we can set up equations using the first few terms of the sequence:
For \( n = 1 \): \( T_1 = a(1)^2 + b(1) + c = 2 \),
For \( n = 2 \): \( T_2 = a(2)^2 + b(2) + c = 10 \),
For \( n = 3 \): \( T_3 = a(3)^2 + b(3) + c = 26 \).

This gives us the system of equations:
\( a + b + c = 2 \),
\( 4a + 2b + c = 10 \),
\( 9a + 3b + c = 26 \).

By solving this system, we find \( a = 2 \), \( b = 1 \), and \( c = -1 \). However, since the sequence starts at 2, we adjust \( c \) to fit the first term correctly, leading to the final formula \( T_n = 2n^2 + n \).