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

Progressions are sequences that follow a common pattern in their terms. We can obtain the formula for the n^th or the general term and for the sum till the n^th term if we find the pattern in any sequence.

Answer: The formula for the general term of the sequence: 2, 6, 12, 20, 30... is \(a_{n}\) = n² + n.

Let's understand the solution in detail.

Explanation:

Given expression: 2, 6, 12, 20, 30, ...

From the given series, we see that:

⇒ 2 + 4 = 6

⇒ 6 + 6 = 12

⇒ 12 + 8 = 20

⇒ 20 + 10 = 30

Hence, we see that the difference between the consecutive terms increases by 2 as the series progresses.

Hence, the first difference is in arithmetic progression, but the second difference is constant here.

Hence, the general term must be quadratic.

Now, we use the general quadratic equation an² + bn + c, where n is the variable expressing the number of terms.

Hence, from the given data:

⇒ When n = 1; a₁ = a + b + c = 2  (equation 1)

⇒ When n = 2; a₁ = 4a + 2b + c = 6 (equation 2)

⇒ When n = 3; a₁ = 9a + 3b + c = 12 (equation 3)

Now, subtracting equation 1 from equation 2, we get:

⇒ 3a + b = 4  (equation 4)

Now, subracting equation 2 from equation 3, we get:

⇒ 5a + b = 6 (equation 5)

Now, we will subtract equation 4 from equation 5.

⇒ 5a - 3a  = 6 - 4

⇒ 2a = 2

⇒ a = 1

From the value of a, we get b = 1, from equation 4 and equation 5.

Now, substituting the values of a and b in equation 1, we get:

⇒ a₁ = (1) + (1) + c = 2

⇒ c = 0

Hence, putting the values of a, b and c in the general equation above, we get (1) n² + (1) n + (0) = n² + n.

Hence, The formula for the general term of the sequence: 2, 6, 12, 20, 30, ... is a_n = n² + n.