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Help how to do this

 

Cai writes down the list of positive integers, excluding squares and cubes and all perfect powers.  His sequence starts
2, 3, 5, 6, 7, 10, 11, ...
What is the 100th term in Cai's list?

 Mar 5, 2023
 #1
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We can start by listing out the first few terms of the sequence and looking for a pattern:

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...

Notice that the sequence includes all positive integers, except for those that are perfect squares, perfect cubes, or any other perfect power (a number that can be written as a positive integer raised to some positive integer power).

To find the 100th term in the sequence, we can continue listing out terms until we get to the 100th term. However, there is a more efficient way to do this.

Let's count how many terms are in the sequence that are less than or equal to some integer n. We can do this by counting the number of perfect squares, perfect cubes, and perfect fourth powers that are less than or equal to n, and subtracting this from n. For example, if n = 20, then the perfect squares less than or equal to 20 are 1, 4, 9, 16, and the perfect cubes less than or equal to 20 are 1, 8. The perfect fourth powers less than or equal to 20 are just 1. So the number of terms in the sequence less than or equal to 20 is:

20 - 5 - 2 - 1 = 12

Therefore, the 12th term in the sequence is the largest term less than or equal to 20.

Now we can use this method to find the 100th term in the sequence. We want to find the largest integer n such that there are 100 terms in the sequence less than or equal to n. Let's try some values of n:

For n = 10, there are:

10 - 3 - 2 - 1 = 4

terms in the sequence less than or equal to 10.

For n = 20, there are:

20 - 5 - 2 - 1 = 12

terms in the sequence less than or equal to 20.

For n = 30, there are:

30 - 5 - 3 - 1 = 21

terms in the sequence less than or equal to 30.

For n = 40, there are:

40 - 6 - 3 - 1 = 30

terms in the sequence less than or equal to 40.

For n = 50, there are:

50 - 7 - 3 - 1 = 39

terms in the sequence less than or equal to 50.

For n = 60, there are:

60 - 7 - 3 - 1 = 49

terms in the sequence less than or equal to 60.

For n = 70, there are:

70 - 8 - 3 - 1 = 58

terms in the sequence less than or equal to 70.

For n = 80, there are:

80 - 8 - 3 - 1 = 68

terms in the sequence less than or equal to 80.

For n = 90, there are:

90 - 9 - 3 - 1 = 77

terms in the sequence less than or equal to 90.

For n = 100, there are:

100 - 9 - 3 - 1 = 87

terms in the sequence less than or equal to 100.

Therefore, the 100th term in the sequence is the largest term less than or equal to 100, which is 97.

So the 100th term in Cai's list is 97.

 Mar 5, 2023

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