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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.

    Equation I
    \(\begin{align}
    t_1 &= ? \\
    d &= ? \\
    S_5 &= 450 \\
    n &= 5 \\
    \end{align}\)
    \[\begin{align}
     S_n &= \frac{n}{2}\left[ {2t_1 + (n - 1)d} \right] \\
     450 &= \frac{5}{2}\left[ {2t_1 + \left( {5 - 1} \right)d} \right] \\
     180 &= 2t_1 + 4d \\
     \end{align}\]
    Equation II
    \(\begin{align}
    t_1 &= ? \\
    d &= ? \\
    S_8 &= 600 \\
    n &= 8 \\
    \end{align}\)
    \[\begin{align}
     S_n &= \frac{n}{2}\left[ {2t_1 + (n - 1)d} \right] \\
     600 &= \frac{8}{2}\left[ {2t_1 + \left( {8 - 1} \right)d} \right] \\
     150 &= 2t_1 + 7d \\
     \end{align}\]

    Subtract Equation I from Equation II.
    \[\begin{align}
    150 &= \cancel{2t_1} + 7d \\ 
    - (180 &= \cancel{2t_1} + 4d)  \\
    \hline {} \\
    - 30 &= \qquad \enspace 3d \\
    - 10 &= \qquad \enspace \enspace d \\
     \end{align}\]
    Luella is actually putting less money in each week, \($10\) less!

  2. Calculate how much Luella put into her savings account the first week.

    Using one of the equations from part a, determine \(t_1 \).
    \(\begin{align}
     150 &= 2t_1 + 7d \\
     150 &= 2t_1 + 7\left( { -10} \right) \\
     220 &= 2t_1  \\
     110 &= t_1
     \end{align}\)
    Luella started with \($110\) in her savings account.

  3. How much money will be in the account if Luella maintains her savings plan for \(20\) weeks?

    \(\begin{align}
     t_1 &= 110 \\
     n &= 20 \\
     d &= -10 \\
     S_{20} &= ? \\
     \end{align}\)
    \[\begin{align}
     S_n &= \frac{n}{2}\left[ {2t_1 + (n - 1)d} \right] \\
     S_{20} &= \frac{{20}}{2}\left[ {2\left( {110} \right) + \left( {20 - 1} \right)\left( { -10} \right)} \right] \\
     S_{20} &= 10\left( {220 - 190} \right) \\
     S_{20} &= 10\left( {30} \right) \\
     S_{20} &= 300 \\
     \end{align}\]

    How can this be less than week five? Recall that the value of \(d \) is \(–10\), therefore Luella is putting in less and less money into the account. At some point, she is actually taking money out of the account! If you calculate the value of the \(20^{th}\) term, you will see that this week, Luella only took money out of the account – not a great savings plan.

    \(\begin{align}
     t_n &= t_1 + (n - 1)d \\
     t_{20} &= 110 + (20 - 1)( -10) \\
     t_{20} &= 110 + (19)( -10) \\
     t_{20} &= 110 - 190 \\
     t_{20} &= -80 \\ 
    \end{align}\)

  4. What assumption do you make to answer part c?

    The assumption is that Luella continues with her savings plan. Another assumption is that Luella continues to save \($10\) less than the previous week until she gets to \($0\), at which time she starts taking money out of the account.


 For further information about arithmetic series, please read through pp. 26 - 27 of Pre-Calculus 11.