What is the sum of the odd integers from 35 to 85, inclusive?

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The sum of the odd integers from 35 to 85 inclusive is 1,560. We first note the key features of our sequence: the first term, the last term, and the constant gap (step) between consecutive odd numbers. a_1 = 35 a_n = 85 step = 2 Next, we count how many terms appear in this evenly spaced list. By the "Counting Evenly Spaced Integers" rule, we subtract the endpoints, divide by the step, and add one because the list is inclusive. n = 85 - 35/2 + 1 = 50/2 + 1 = 25 + 1 = 26 For an evenly spaced list, the average value equals the mean of the first and last terms. average = 35 + 85/2 = 120/2 = 60 Finally, we apply the "Sum of Evenly Spaced Integers" rule: the sum equals the average multiplied by the number of terms. sum = 60 26 = (60 20) + (60 6) = 1200 + 360 = 1560

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$We first note the key features of our sequence: the first term, the last term, and the constant gap (step) between consecutive odd numbers.$	$a_{1} = 35 a_{n} = 85 step = 2$
$Next, we count how many terms appear in this evenly spaced list. By the "Counting Evenly Spaced Integers" rule, we subtract the endpoints, divide by the step, and add one because the list is inclusive.$	$n = \frac{85 - 35}{2} + 1 = \frac{50}{2} + 1 = 25 + 1 = 26$
$For an evenly spaced list, the average value equals the mean of the first and last terms.$	$average = \frac{35 + 85}{2} = \frac{120}{2} = 60$
$Finally, we apply the "Sum of Evenly Spaced Integers" rule: the sum equals the average multiplied by the number of terms.$	$sum = 60 \times 26 = (60 \times 20) + (60 \times 6) = 1200 + 360 = 1560$

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