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Find the sum of the positive terms of the arithmetic sequence
85,78,71,dots

Find the sum of the positive terms of the arithmetic sequence $85,78,71, \ldots$

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Identify and classify polygons

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Q. Find the sum of the positive terms of the arithmetic sequence

85,78,71, \ldots

Identify Decreasing Pattern: The sequence is decreasing by $7$ each time ( $85 - 78 = 7$ ). To find the sum of the positive terms, we need to find the last positive term in the sequence.
Find Last Positive Term: The $n$ th term of an arithmetic sequence is given by $a_n = a_1 + (n - 1)d$ , where $a_1$ is the first term and $d$ is the common difference. We need to solve for $n$ when $a_n > 0$ .
Set Up Inequality: Let's set up the inequality: $85 + (n - 1)(-7) > 0$ .
Solve Inequality: Solving the inequality: $85 - 7n + 7 > 0$ .
Simplify Inequality: Simplify the inequality: $92 - 7n > 0$ .
Calculate $n$ Value: Divide by $-7$ and reverse the inequality sign: $n < \frac{92}{7}$ .
Find $13$ th Term: Calculate $n$ : $n < 13.14$ . Since $n$ must be a whole number, the last positive term is when $n = 13$ .
Use Arithmetic Series Formula: Now we find the $13^{\text{th}}$ term: $85 + (13 - 1)(-7) = 85 - 84 = 1$ .
Substitute Values: The sum of an arithmetic series is given by $S_n = \frac{n}{2} * (a_1 + a_n)$ . We'll use this formula to find the sum of the first $13$ terms.
Calculate Sum: Substitute the values into the formula: $S_{13} = \frac{13}{2} \times (85 + 1)$ .
Calculate Sum: Substitute the values into the formula: $S_{13} = \frac{13}{2} \times (85 + 1)$ .Calculate the sum: $S_{13} = 6.5 \times 86$ .
Calculate Sum: Substitute the values into the formula: $S_{13} = \frac{13}{2} \times (85 + 1)$ .Calculate the sum: $S_{13} = 6.5 \times 86$ .Final calculation: $S_{13} = 559$ .

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