Answers:
Geometric:
- 1, 2, 4, 8, ...
- 250, 50, 10, 2, ...
- 2, -6, 18, -54, ...
Arithmetic:
- -3, -6, -9, -12, ...
- 50, 100, 150, 200, ...
- 1, 3, 5, 7, ...
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Explanation:
A geometric sequence is one where we generate terms by multiplying each term by the same number to get the next number.
In the case of {1,2,4,8,...} we multiply each term by 2.
- first term = 1
- second term = 2*(first term) = 2*1 = 2
- third term = 2*(second term) = 2*2 = 4
- fourth term = 2*(third term) = 2*4 = 8
And so on. We say the common ratio is 2.
For the geometric sequence {250, 50, 10, 2, ...} the common ratio is 1/5 since 250*1/5 = 50 and 50*1/5 = 10, etc.
For the geometric sequence {2, -6, 18, -54, ...} we are multiplying each term by -3.
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Arithmetic sequences are similar to geometric ones. Instead of multiplying, we add on the same thing each time to get new terms.
The sequence {-3, -6, -9, -12, ...} is arithmetic because we are adding on -3 each time
- first term = -3
- second term = (first term) + (-3) = (-3)+(-3) = -6
- third term = (second term) + (-3) = (-6)+(-3) = -9
- fourth term = (third term) + (-3) = (-9)+(-3) = -12
This process goes on for as long as you want. We say the common difference is -3
For the arithmetic sequence {50, 100, 150, 200, ...} the common difference is 50 because we are adding 50 on each time to get a new term.
The arithmetic sequence {1, 3, 5, 7, ...} has the common difference 2.
