Use inductive reasoning to describe each pattern, then find the next two numbers in each pattern.
1,1/4,1/9,and 1/16... those are fractions except for the number 1 in front.
Find the 2nd,5th, and 9th terms of each sequence.
1/2,1/3,1/6,and 0... those are fractions except for the number 0.
To describe the first pattern, let's analyze the relationship between the consecutive terms:
1 = 1/1^2
1/4 = 1/2^2
1/9 = 1/3^2
1/16 = 1/4^2
We can observe that the pattern is formed by taking the reciprocal of the series of square numbers. The next term would be the reciprocal of 5^2, which is 1/25, and the following term would be the reciprocal of 6^2, which is 1/36.
So, the next two numbers in this pattern are 1/25 and 1/36.
Now let's analyze the second pattern:
1/2 = 1/2
1/3 = 1/3
1/6 = 1/2 * 1/3
0 = 1/2 * 0
We can observe that the pattern is formed by dividing 1 by a series of consecutive numbers. The second term in the pattern is 1/3, the fifth term is 1/6, and the ninth term is 1/10.
So, the second, fifth, and ninth terms of this sequence are 1/3, 1/6, and 1/10.