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Printjmc
algebra senior
Problem
The sum of the first 5 terms of an arithmetic series is . The sum of the first 10 terms of this arithmetic series is . What is the first term of the series?
Solution
Let the first term be and the common difference be . The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms. The fifth term is , so the sum of the first five terms is which implies that , so .
The tenth term is , so the sum of the first ten terms is which implies that , so .
From the equation , , and from the equation , , so Then , so .
The tenth term is , so the sum of the first ten terms is which implies that , so .
From the equation , , and from the equation , , so Then , so .
Final answer
\frac{42}{5}