Find a formula for the general term of the sequence 2, 6, 12, 20, 30

The given sequence is 2, 6, 12, 20, 30. To find a formula for the general term, let’s first look for a pattern in the sequence.

If we examine the terms:

• 1st term: 2
• 2nd term: 6
• 3rd term: 12
• 4th term: 20
• 5th term: 30

Next, let’s look at the differences between consecutive terms:

• 6 – 2 = 4
• 12 – 6 = 6
• 20 – 12 = 8
• 30 – 20 = 10

So, the first differences are 4, 6, 8, 10. Now, let’s look at the differences of these differences:

• 6 – 4 = 2
• 8 – 6 = 2
• 10 – 8 = 2

The second differences are constant (equal to 2), which suggests that the original sequence can be represented by a quadratic function.

Generally, a quadratic function can be expressed as:

an^2 + bn + c

To find the coefficients a, b, and c, we can set up a system of equations using the known terms of the sequence:

• For n = 1: a(1)^2 + b(1) + c = 2
• For n = 2: a(2)^2 + b(2) + c = 6
• For n = 3: a(3)^2 + b(3) + c = 12

This gives us the following equations:

1. a + b + c = 2
2. 4a + 2b + c = 6
3. 9a + 3b + c = 12

By solving this system, we can determine the values of a, b, and c. After some calculations, we find:

• a = 1
• b = 1
• c = 0

This tells us that the general formula for the nth term of the sequence is:

T(n) = n^2 + n

Therefore, for any term n, you can compute it using this formula. For example:

• T(1) = 1^2 + 1 = 2
• T(2) = 2^2 + 2 = 6
• T(3) = 3^2 + 3 = 12
• T(4) = 4^2 + 4 = 20
• T(5) = 5^2 + 5 = 30

This matches the original sequence, confirming our formula is correct.