Question

Determine if the sequence 4, 10, 19, 31,... is arithmetic. If it is, determine the first term a and the common difference d

Answer 1

It is not arithmetic, so therefore there is no common difference

Explanation:

An arithmetic sequence is a sequence where each term increases by adding/subtracting some constant k. There is no constant term in the above pattern.

Answer 2

not an arithmetic sequence

Explanation:

An arithmetic sequence is a sequence of numbers that have a common difference. That is, the difference between any term and the previous term is constant for the sequence. Other kinds of sequences have other relationships between the differences.

Differences

The "first" differences of this sequence are ...

• 10-4 = 6
• 19-10 = 9
• 31 -19 = 12

The first differences are not constant. However, we notice the "second" differences are constant. These are the differences of successive first differences.

• 9 -6 = 3
• 12 -9 = 3 . . . . . . constant 2nd differences

Sequence type

The first differences are not constant, so this sequence is not an arithmetic sequence.

For polynomial sequences, the level of constant difference tell you the degree of the polynomial describing the sequence. This sequence has constant 2nd-level differences, so can be described by a 2nd degree (quadratic) polynomial:

f(n) = 1.5n² +1.5n +1 . . . . . a quadratic sequence

__

Additional comment

Sequences that are exponential have differences that have a common ratio. That ratio is the same at every level. It is the base of the exponential function.