Determine the rule for the number pattern and use it to find the missing values.

Which of the following statements are true about the pattern above?
Select all that are true.
3,7,15 , , , ,....
The 10th term in the pattern will be 30.
Each term in the pattern will be a multiple of 3.
Each term in the pattern will be prime.
Each term in the pattern will be odd.

9514 1404 393

Answer:

  Each term in the pattern will be odd.

Step-by-step explanation:

The first differences are ...

  7 -3 = 4

  15 -7 = 8

These differ by 4, and the second is double the first.

These relationships between the first differences give rise to two possible sequences: a) an exponential sequence; b) a quadratic sequence. We can use a graphing calculator to find the coefficients of each of the patterns.

Exponential Sequence

  a[n] = 2·2^n -1

  The sequence is 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047.

  The 10th term is 2047.

Quadratic Sequence

  a[n] = 2n^2 -2n +3

  The sequence is 3, 7, 15, 27, 43, 63, 87, 115, 147, 183.

  The 10th term is 183.

__

Based on the above, the only thing we can say about these sequences is that they are all odd numbers.

Clearly, the 10th term is not 30. 7 is not a multiple of 3, so that observation fails immediately. 15 is not prime, so that observation also fails immediately.

_____

Additional comment

The differences of first differences are called "second differences". The differences of those are "third differences". and so on. If you keep taking differences of a sequence described by a polynomial, the n-th differences being constant means the sequence is described by an n-th degree polynomial.

Here, if we assume the 2nd differences are constant at 4, then the sequence is described by a 2nd-degree polynomial. (Similarly, constant first-differences are described by a 1st-degree polynomial--a linear function.)

If sequential levels of differences are all exponential sequences, then the sequence itself is an exponential sequence. As here, it may have a vertical offset. The base of the exponential will be the common ratio of the differences. (Here, the first differences have a ratio of 8/4 = 2, so the sequence could be exponential with a base of 2.)

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