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Find the sum of the finite arithmetic series. $\sum_{n=1}^{10} (7n+4)$ $\newline$ ______

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Find the sum of an arithmetic series

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Q. Find the sum of the finite arithmetic series.

\sum_{n=1}^{10} (7n+4)

\newline

______

Arithmetic Series Sum Formula: To find the sum of an arithmetic series, we can use the formula $S_n = \frac{n}{2} * (a_1 + a_n)$ , where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the $n$ th term. In this case, we need to find the sum of the first $10$ terms of the series where each term is given by $7n+4$ .
Find First Term: First, we need to find the first term $a_1$ by plugging $n=1$ into the formula $7n+4$ . This gives us $a_1 = 7(1) + 4 = 11$ .
Find Tenth Term: Next, we need to find the tenth term $a_{10}$ by plugging $n=10$ into the formula $7n+4$ . This gives us $a_{10} = 7(10) + 4 = 70 + 4 = 74$ .
Calculate Sum: Now that we have the first term $a_1$ and the tenth term $a_{10}$ , we can use the sum formula $S_n = \frac{n}{2} * (a_1 + a_n)$ . Plugging in the values, we get $S_{10} = \frac{10}{2} * (11 + 74)$ .
Final Result: Simplifying the expression, we get $S_{10} = 5 \times (11 + 74) = 5 \times 85 = 425$ .

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