# This is really tricking me, please help in any way possible. Choose the appropriate category for each of the following sequences (arithmetic, geometric, or neither). Give the common differences or...

Choose the appropriate category for each of the following sequences (arithmetic, geometric, or neither). Give the common differences or common ratio, or choose "not applicable." Find the tenth term (use applicable formulas).

a.) 1, -1, 1, -1, ....

Choose: arithmetic or geometric or neither

Common Difference= _______

Common Ratio= _______

Or not applicable

10th term: ______

b.) 1, 1, 2, 3, 5, 8, 13, ...

Choose: arithmetic or geometric or neither

Common Difference= _______

Common Ratio= _______

Or not applicable

10th term: ______

c.) 11, 9, 7, 5, ...

Choose: arithmetic or geometric or neither

Common Difference= _______

Common Ratio= _______

Or not applicable

10th term: ______

ishpiro | Certified Educator

Recall the definitions of the arithmetic and geometric sequences in order to answer these questions:

In an arithmetic sequence, each term is obtained by adding a number to the previous term. The number added is always the same and called a common difference. It can be shown that the nth term of the arithmetic sequence is

`a_n = a_1 + (n-1)d` , where d is common difference and a_1 is the first term.

In a geometric sequence, each term is obtained by multiplying the previous term by a number. This number is always the same and called a common ratio. It can be shown that the nth term of the geometric sequence is

`b_n = b_1r^(n-1)` , where r is the common ratio and b_1 is the first term.

So, check if the given sequences satisfy the definition of either sequence:

In a), if you divide the second term by first, then the third term by second, and so on, you will notice that the ratio is always going to be the same, -1. So this is a geometric sequence with the common ratio r = -1. Then, its 10th term will be

`b_10 = b_1r^(10-1) = 1*(-1)^(9) = -1`

In b), notice that the ratios won't be the same:

1/1 = 1, 2/1 = 2.

Nor the differences between the consecutive terms are the same:

1 -1 = 0, 2 - 1 = 1, 3 - 2 = 1,  5 - 3 = 2

So this sequence is neither arithmetic nor geometric. However, notice that each term is the sum of the two previous terms: ( 2 = 1+ 1, 3 = 2+1, 5 = 3  + 2, and so on.) This is a famous Fibonacci sequence. ("Google" it to learn about its many interesting properties.) We already have seven terms, so we can continue the pattern to find the 10th term:

8th term: 13 + 8 = 21

9th term 21 + 13 = 34

10th term: 34 + 21 = 55

In c) , the difference between the consecutive terms is the same:

9 - 11 = -2, 7 - 9 = -2, 5 - 7 = -2

So this is an arithmetic sequence with the common difference d = -2.

Its 10th term will be

`a_10 = a_1 + d(10 - 1) = 11 + (-2)(10-1) = 11-18 = -7`