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.
The denominator increases by one (which is squared) each time.
1^2 = 1
2^2 = 4
and so on.
The second term (of course) is 1/3, but the zero has me baffled.... Is there a typo?
Sorry that I can't be of more help.
To describe the patterns and find the next two numbers in each sequence, we can use inductive reasoning. Let's analyze each pattern separately:
Pattern 1: 1, 1/4, 1/9, 1/16...
In this pattern, we can observe that each number is the reciprocal of a perfect square. The first number, 1, can be expressed as 1/1^2. The second number, 1/4, can be expressed as 1/2^2. Similarly, the third number, 1/9, can be expressed as 1/3^2, and the fourth number, 1/16, can be expressed as 1/4^2. Therefore, the pattern continues with the reciprocal of the next perfect square, which is 1/5^2 or 1/25. Following this pattern, the next two numbers would be 1/6^2 or 1/36, and 1/7^2 or 1/49.
Therefore, the next two numbers in the pattern are 1/25, 1/36, followed by 1/49.
Pattern 2: 1/2, 1/3, 1/6, 0...
In this pattern, we can observe that each number is a fraction where the numerator is always 1, and the denominator is a multiple of 2. The first number, 1/2, has a denominator of 2. The second number, 1/3, has a denominator of 3. The third number, 1/6, has a denominator of 6. Finally, the fourth number is 0.
To find the 2nd, 5th, and 9th terms of this sequence, we can list out the first few terms:
1/2, 1/3, 1/6, 0, ...
The 2nd term is 1/3, the 5th term is 0, and the 9th term can be found by noticing that the pattern repeats after every third term. Therefore, we can look at the 9th term as the same as the 3rd term, which is 1/6.
Therefore, the 2nd, 5th, and 9th terms of the sequence are 1/3, 0, and 1/6, respectively.