List the terms that complete a possible pattern in each of the following and state whether the pattern is arithmetic, geometric, or neither:

(a) 38, 33, 28, 23, 18, …

(b) 640, 320, 160, 80, …

(e) 1, ___, ___, ___ 25, 36, 49

http://www.jiskha.com/display.cgi?id=1310602681

a)The pattern is continued by subtracting 5 each time.

b)The pattern is continued by dividing by 2 each time.

e)

1,4,9,16,25,36,49

In this series,all numbers are the squares of consecutive integers.

1^2=1

2^2=4

3^2=9

4^2=16

5^2=25

6^2=36

7^2=49

To determine the pattern in each of the given sequences, we need to analyze the differences between consecutive terms.

(a) 38, 33, 28, 23, 18, ...

To find the common difference, we subtract each term from its succeeding term:
33 - 38 = -5
28 - 33 = -5
23 - 28 = -5
18 - 23 = -5

We can see that each term is decreasing by 5. Therefore, the pattern is arithmetic. To determine the missing terms, we continue subtracting 5 from the previous term:

18 - 5 = 13
13 - 5 = 8

So, the completed pattern is 38, 33, 28, 23, 18, 13, 8.

(b) 640, 320, 160, 80, ...

Similarly, we find the common ratio by dividing each term by its preceding term:
320 ÷ 640 = 0.5
160 ÷ 320 = 0.5
80 ÷ 160 = 0.5

By observing that each term is being divided by 2, we can conclude that the pattern is geometric. To determine the missing term, we continue dividing the previous term by 2:

80 ÷ 2 = 40

So, the completed pattern is 640, 320, 160, 80, 40.

(e) 1, ___, ___, ___ 25, 36, 49

In this case, the pattern is not clearly arithmetic or geometric. We can observe that the given terms after 1 are perfect squares (5², 6², 7²) and that the differences between consecutive terms seem to increase by 1 each time.

To determine the missing terms, we need to continue the pattern. The next perfect square after 7² (49) is 8²:

8² = 64

So, the completed pattern is 1, 64, 81, 100, 25, 36, 49.