How do you find the nth term of 5, 15, 35, 65?

The nth term of the sequence 5, 15, 35, 65 is given by the formula \( n^3 + 4n - 1 \).

To find the nth term of a sequence, we first need to identify the pattern or rule that the sequence follows. Let's analyse the given sequence: 5, 15, 35, 65.

First, we calculate the differences between consecutive terms:
- 15 - 5 = 10
- 35 - 15 = 20
- 65 - 35 = 30

Next, we calculate the second differences:
- 20 - 10 = 10
- 30 - 20 = 10

Since the second differences are constant, this indicates that the sequence is quadratic. However, the differences suggest a cubic term might be involved. To find the exact formula, we assume it is of the form \( an^3 + bn^2 + cn + d \).

We set up equations using the first few terms:
1. For n = 1: \( a(1)^3 + b(1)^2 + c(1) + d = 5 \)
2. For n = 2: \( a(2)^3 + b(2)^2 + c(2) + d = 15 \)
3. For n = 3: \( a(3)^3 + b(3)^2 + c(3) + d = 35 \)
4. For n = 4: \( a(4)^3 + b(4)^2 + c(4) + d = 65 \)

Solving these equations, we find that \( a = 1 \), \( b = 0 \), \( c = 4 \), and \( d = -1 \). Therefore, the nth term formula is \( n^3 + 4n - 1 \).

This formula allows us to find any term in the sequence by substituting the value of n. For example, for the 5th term, substitute n = 5 into the formula: \( 5^3 + 4(5) - 1 = 125 + 20 - 1 = 144 \).