Arithmetic
Arithmetic
Series Formula
Example 2
The sum of the first 50 terms of an arithmetic sequence is 4,800. If the 50th term is 194, find the first term and the common difference.
and
![SeriesFormulaEx2-9 SeriesFormulaEx2-9](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-9.png)
Step 1. Determine a1 and an.
![SeriesFormulaEx2-1 SeriesFormulaEx2-1](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-1.png)
![SeriesFormulaEx2-2 SeriesFormulaEx2-2](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-2.png)
Step 2. Determine the number of pairs that will make this sum.
![SeriesFormulaEx2-3 SeriesFormulaEx2-3](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-3.png)
Step 3. Substitute into the formula and solve.
![SeriesFormulaEx2-4 SeriesFormulaEx2-4](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-4.png)
![SeriesFormulaEx2-5 SeriesFormulaEx2-5](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-5.png)
![SeriesFormulaEx2-6 SeriesFormulaEx2-6](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-6.png)
The first term, a1, is -2.
Step 4. Determine the formula for the sequence.
![SeriesFormulaEx2-7 SeriesFormulaEx2-7](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-7.png)
![SeriesFormulaEx2-8 SeriesFormulaEx2-8](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-8.png)
Step 5. Solve for d.
![SeriesFormulaEx2-9 SeriesFormulaEx2-9](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-9.png)
![SeriesFormulaEx2-10 SeriesFormulaEx2-10](https://claregladwinresd.glk12.org/pluginfile.php/1011/mod_book/chapter/597/Exponential_and_Logarithmic_Functions/Images/SeriesFormulaEx2-10.png)
The common difference is 4.