Tell whether the sequence 1/3,0,1,-2... I'd arithmetic, geometric,or neither. Find the next three terms of the sequence
1/3 to 0 is - 1/3
0 to 1 is one
so it is not an arithmetic sequence which is constant difference
1/3 * 0 = 0
0 * nothing I know of = 1
so it is not a geometric sequence
now let's see if we can find some other pattern
1/3 to 0 = - 1/3
0 to 1 = +1
1 to - 2 = -3
maybe
difference = -3 times previous difference?
-2 -3(-3) = -2+9 = 7
7 -3(9) = 7 - 27 = -20
-20 -3(-27) = 61
SO
1/3, 0 , 1 , -2 , 7, -20 , 61
To determine whether the given sequence is arithmetic, geometric, or neither, we need to examine the differences between consecutive terms.
Let's calculate the differences between consecutive terms:
1/3 - 0 = 1/3
0 - 1/3 = -1/3
1 - 0 = 1
-2 - 1 = -3
As we can see, the differences between consecutive terms are not constant. For example, the difference between the first and second terms is 1/3, but the difference between the second and third terms is -1/3. Therefore, the given sequence is not arithmetic.
Now, let's check if the sequence is geometric by calculating the ratios between consecutive terms:
(0)/(1/3) = 0
(1)/(0) = undefined
(-2)/(1) = -2
Again, we can see that the ratios between consecutive terms are not constant. Additionally, one of the ratios is undefined, indicating that the sequence is not geometric either.
Therefore, the given sequence is neither arithmetic nor geometric.
To find the next three terms in the sequence, we can continue the pattern that we observe:
The pattern so far: 1/3, 0, 1, -2.
Continuing the pattern, we can add the terms:
-2 + 3 = 1
1 + 3 = 4
4 + 3 = 7
Hence, the next three terms of the sequence are: 1, 4, 7.
To determine whether a sequence is arithmetic, geometric, or neither, we need to look for patterns in the differences or ratios between consecutive terms.
In this case, let's find the differences between consecutive terms:
1/3 - 0 = 1/3
0 - 1 = -1
1 - 0 = 1
-2 - 1 = -3
As we can see, the differences are not constant. An arithmetic sequence has a constant difference between terms, so we can conclude that this sequence is not arithmetic.
Now, let's find the ratios between consecutive terms:
1/3 ÷ 0 = undefined (division by zero)
0 ÷ 1 = 0
1 ÷ 0 = undefined
-2 ÷ 1 = -2
Again, the ratios are not constant, and there are undefined values. A geometric sequence has a constant ratio between terms, so we can also conclude that this sequence is not geometric.
Therefore, the sequence 1/3, 0, 1, -2... is neither arithmetic nor geometric.
To find the next three terms of the sequence, we can continue the pattern observed in the differences or ratios:
If we continue the pattern of differences (1/3, -1, 1, -3), the next difference would be -5. So, the next three terms would be:
-2 - 5 = -7
-7 - (-2) = -5
Therefore, the next three terms of the sequence are: -2, -7, -5.