Sure! Let's break down the solution step-by-step to find the first 5 elements of set H, which are numbers that are both divisible by 4 (set G) and perfect squares (set F).
1. Identify Set G (Positive integers divisible by 4):
- These are the numbers that can be written as [tex]\( 4n \)[/tex] where [tex]\( n \)[/tex] is a positive integer.
- The sequence is: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, ...
For clarity, let's list the first 20 elements:
[tex]\[
\{4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80\}
\][/tex]
2. Identify Set F (Perfect squares):
- These are numbers that can be written as [tex]\( n^2 \)[/tex] where [tex]\( n \)[/tex] is a positive integer.
- The sequence is: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, ...
For clarity, let's list the first 20 elements:
[tex]\[
\{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400\}
\][/tex]
3. Find the common elements:
- The elements that are in both G and F are: 4, 16, 36, 64, 100...
- So, set H is: 4, 16, 36, 64, 100, 144, 196, 256, 324, ...
4. List the first 5 elements of set H:
- The first 5 common elements are: 4, 16, 36, 64, 100
Therefore, the first 5 elements of set H, which are both divisible by 4 and perfect squares, are:
[tex]\[
\{4, 16, 36, 64, 100\}
\][/tex]
