The pattern in the given series of amount in the account are in the form of
arithmetic and geometric progression.
- The function for Option 1 is; [tex]\underline{ f(n) = 1,100 + (n - 1) \cdot 100}[/tex]
- The function for Option 2 is; [tex]\underline{f(n) = 1,100 \times 1.1^{(n - 1)}}[/tex]
Reasons:
The given table of values is presented as follows;
[tex]\begin{tabular}{c|c|c|c|}Number of years&1&2&3\\Option 1 (Amount in dollars)&1,100&1,200&1,300\\Option 2 (Amount in dollars)&1,100&1,210&1,331\end{array}\right][/tex]
In Option 1, the amount in dollars for each year has a common difference of d = 100
The first term, a = 1,100
Therefore;
The Option 1 can be represented as an arithmetic progression , A.P. in the
form, tₙ = a + (n - 1)·d as follows;
For the Option 1, we have;
- The amount in dollars after n years, [tex]\underline{ f(n) = 1,100 + (n - 1) \cdot 100}[/tex]
For Option 2, it is possible to find;
1,331 ÷ 1,210 = 1,210 ÷ 1,100 = 1.1
Therefore;
The terms in the Option 2 have a common ratio of r = 1.1
The Option 2 is a geometric progression, G.P.
The first term in Option 2 is a = 1,100
Which gives, the nth term, tₙ = a·r⁽ⁿ ⁻ ¹⁾
Therefor;
- The function for the Option 2 is; [tex]\underline{f(n) = 1,100 \times 1.1^{(n - 1)}}[/tex]
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