Given that:
- Cory saves $30 in June.
- Each month he plans to save 10% more than the previous month.
- He saves money from June to December.
You can convert 10% to a decimal number by dividing it by 100:
[tex]10\text{ \%}=\frac{10}{100}=0.1[/tex]
You already know that he has $30 in June. Then, you can determine that the amount of money (in dollars) he will save in July is:
[tex]30+(30\cdot0.1)=33[/tex]
By definition, the formula for the Sum of a Geometric Sequence is:
[tex]S_n=\frac{a_1(1-r^n)}{1-r}[/tex]
Where "r" is the Common Ratio, "n" is the number of terms, and this is the first term:
[tex]a_1[/tex]
In this case, you can identify that the first term is:
[tex]a_1=30[/tex]
And the second term is:
[tex]a_2=33[/tex]
Therefore, you can find the Common Ratio as follows:
[tex]r=\frac{33}{30}=1.1[/tex]
Since he saves money from June to December, the sequence has 7 terms. Then:
[tex]n=7[/tex]
Now you can substitute values into the formula and evaluate:
[tex]S_7=\frac{30(1-(1.1)^7)}{1-r}\approx284.615[/tex]
Hence, the answer is:
[tex]\text{ \$}284.615[/tex]
