This can be done using combination of page styles AND manual page breaks.
Many users don't really understand how page styles works, so confusion is made... First of all when you select page style there is always a "next page style" definition. So for example "First Page" style has a definition of next style as "Default Style". So if your cursor is on first page and you select "First Page" style then AUTOMATICALLY second page gets "Default Style". If you would like to set a different style for second page, you can't just select style for second page, because style applied to first page AUTOMATICALLY sets page style for second page. To get out of this next page style problem, you need to use manual page break and specify the page style for next page. This sounds complicated, but if you look into writing a book when you have a chapter page and normal pages, then when you select chapter page you automatically get normal page style for next page, so not so crazy anymore.
For your case, we need to specify first and sixth page as "First Page" style. (Note: "We can also create new style and name it what ever you like, but 'First Page' style has all we need, so we can just use it.) And for all other pages we will use Default Style. To break out of "next page style" problem described in previous paragraph, we need to put manual page break in the last line of page 5 and select 'First Page' style for next page (page 6).
Step-by-step instructions:
1. Move cursor to the first page.
2. Format | Styles and Formatting.
3. Format window opens. Click on 4th icon with name Page Styles.
4. (because of step 1 you are on first page). From page style list double click on First Page style to apply this style for first page. Don't forget 'First Page' style has a definition of next style as 'Default Style'. Note: You can check this out clicking on First Page style and click on Modify, on Organizer tab there is Next Style definition.
5. In status bar it is displayed "First page" that indicates your current page style. Note: if you don't see this then check if Status Bar is turned on in View | Status Bar.
6. Move cursor to second page. We will just observe on second page, to see if everything is as expected. In status bar there should be Default Style displayed (Do you remember what I wrote about "Next Style" definition of First Page style?). So second page is done like we want, no change is required on second page.
7. Move cursor to the last empty line in the the page 5.
8. Insert | Manual Break. Type = Page break, Style = First page.
9. Insert | Footer. Now there should be three sub-options (All, Default Style, First Page). Select Default Style.
10. Type in some text and this footer will appear only ... (more)