Question

Examine the following sequence 1,4,9, 16, 25 Why is 36 the next number in the sequence? Because the pattern is:

a. Add the next even number
b. Square the number of the next term, n
c. Cube the number of the term, n
d. Add 2 + the number of the term, n

Answer 1

b. Square the number of the next term, n

Explanation:

You want to know the pattern of the sequence 1, 4, 9, 16, 25, ... and why it tells you the next number is 36.

Pattern

Usually we look for a common difference or a common ratio when we're trying to identify a sequence as arithmetic or exponential. This sequence has neither, as the differences are 3, 5, 7, 9, and the ratios are 4, 9/4, 25/16.

Neither of these are constant sequences. However, we do notice that the sequence of differences has a common difference of 2. The fact that "second differences" are common (all the same) means the sequence can be described by a second-degree polynomial.

Looking at the sequence values, we see they are ...

1 = 1², 4 = 2², 9 = 3², 16 = 4², 25 = 5²

so we expect the 6th term to be the square of 6, 6² = 36. This is in line with the predicted next number.

The pattern is ...

square the number of the next term, n