What is the formula for the general term of the sequence 2, 6, 12, 20, 30?

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The given sequence is 2, 6, 12, 20, 30. To find a formula for the general term of this sequence, we can first look at the differences between consecutive terms:

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

Now, these differences are: 4, 6, 8, 10. Next, we can find the differences of these differences:

  • 6 – 4 = 2
  • 8 – 6 = 2
  • 10 – 8 = 2

Here, we can see that the second differences are constant (always 2), which indicates that the original sequence is quadratic. We can express the general term of a quadratic sequence in the form:

T(n) = an2 + bn + c

Where:

  • T(n) is the nth term of the sequence
  • a, b, and c are constants that need to be determined

To find the values of a, b, and c, we can use the first few terms of the sequence:

1. For T(1) = 2:
a(1)2 + b(1) + c = 2
This simplifies to:
a + b + c = 2

2. For T(2) = 6:
a(2)2 + b(2) + c = 6
This simplifies to:
4a + 2b + c = 6

3. For T(3) = 12:
a(3)2 + b(3) + c = 12
This simplifies to:
9a + 3b + c = 12

Now we can rewrite these equations:

  • Equation 1: a + b + c = 2
  • Equation 2: 4a + 2b + c = 6
  • Equation 3: 9a + 3b + c = 12

Next, we can solve this system of equations. From Equation 1, expressing c in terms of a and b gives:

c = 2 – a – b

Substituting c into Equations 2 and 3:

  • Equation 2 becomes: 4a + 2b + (2 – a – b) = 6
    Simplifying yields:
    3a + b = 4 (Equation A)
  • Equation 3 becomes: 9a + 3b + (2 – a – b) = 12
    This simplifies to:
    8a + 2b = 10
    Which gives:
    4a + b = 5 (Equation B)

We can solve Equations A and B as follows:

From Equation A, we can express b in terms of a:

b = 4 – 3a

Substituting b into Equation B:

4a + (4 – 3a) = 5
This simplifies to:
a + 4 = 5
Thus:
a = 1

We can now find b:

b = 4 – 3(1) = 1

Finally, substituting a and b back into the expression for c:

c = 2 – 1 – 1 = 0

Therefore, the values are:

  • a = 1
  • b = 1
  • c = 0

This means the formula for the general term T(n) of the sequence is:

T(n) = n2 + n

In conclusion, the general term for the sequence is:

T(n) = n2 + n

For example:
For n = 1: T(1) = 12 + 1 = 2
For n = 2: T(2) = 22 + 2 = 6
For n = 3: T(3) = 32 + 3 = 12
And so on.

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