Question

Find the formula for the nth term of the sequence: 0,7,18,33,52​

Answer 1

t(n) = 2n² + n - 3

Explanation:

Given sequence:

• 0, 7, 18, 33, 52

Find the formula for nth term t(n)

Solution

Sequence is neither arithmetic nor geometric, analyzing:

• 0, 7, 18, 33, 52

First level difference

• 7, 11, 15, 19

Second level difference

• 4, 4, 4

As there are two levels, the formula should be in the form of:

• t(n) = an² + bn + c

Trying the formula with the known terms:

• t(1) = 0 ⇒ a+b + c= 0

• t(2) = 4a + 2b + c = 7 ⇒ 3a + b = 7

• t(3) = 9a + 3b + c = 18 ⇒ 3(3a+b) + c = 18 ⇒ 21 + c = 18 ⇒ c = -3

Going back to previous equations to find remaining coefficients:

• t(1) ⇒ a+ b = 3 ⇒ t(2) = 2a + (a+b) = 7 ⇒2a + 3 = 7 ⇒ a= 2 ⇒ b = 1

So the formula becomes:

t(n) = 2n² + n - 3