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Luella has \($450\) in her savings account in the fifth week of saving, and \($600\) in the eighth week of saving.

1. Assuming Luella has been adding money each week into her savings account following an arithmetic sequence, determine the amount she increases her deposit by each week.
   
   
2. Calculate how much Luella put into her savings account the first week.
   
   
3. How much money will be in the account if Luella maintains her savings plan for \(20\) weeks?
   
   
4. What assumption do you make to answer part c?