Question

Here are the first four terms of a sequence.

4
11
22
37
Find an expression, in terms of n, for the nth term of this sequence.

Answer 1

Explanation:

Answer 2

• aₙ = 2n² + n + 1

Explanation:

We observe that the difference of terms is 4, 7 and 15 and next level difference is 4. It means the sequence is quadratic.

We can compare this with simple quadratic sequence

• 1, 4, 9, 16

and find out that doubling each term gives us

• 2, 8, 18, 32

This is close to our sequence, write the terms as follows to find exact rule

The first term:

• a₁ = 4 = 2*1² + 2 = 2*1² + 1 + 1

The second term

• a₂ = 11 = 2*2² + 3 = 2*2² + 2 + 1

The third term:

• a₃ = 22 = 2*3² + 4 = 2*3² + 3 + 1

The fourth term:

• a₄ = 37 = 2*4² + 5 = 2*4² + 4 + 1

The nth term as per observation above is:

• aₙ = 2n² + n + 1