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What is the sum of the odd integers from 23 to 75, inclusive?

Explanation	Calculations	Help
$We first note the key features of our sequence: the first term, the last term, and the constant gap (step) between consecutive odd numbers.$	$a_{1} = 23 a_{n} = 75 step = 2$
$Next, we count how many terms appear in this evenly spaced list. By the "Counting Evenly Spaced Integers" rule, we subtract the endpoints, divide by the step, and add one because the list is inclusive.$	$n = \frac{75 - 23}{2} + 1 = \frac{52}{2} + 1 = 26 + 1 = 27$	Theory & Tactics Method Card EVEN1-B Click to view full details
$We then find the average of the sequence, which for an evenly spaced list is the mean of the first and last terms.$	$average = \frac{23 + 75}{2} = \frac{98}{2} = 49$
$Finally, we apply the "Sum of Evenly Spaced Integers" rule: the sum equals the average multiplied by the number of terms.$	$sum = 49 \times 27 = (49 \times 20) + (49 \times 7) = 980 + 343 = 1323$	Theory & Tactics Method Card EVEN1-D Click to view full details

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B

What is the sum of the odd integers from 23 to 75, inclusive?

Explanation	Calculations	Help
$We first note the key features of our sequence: the first term, the last term, and the constant gap (step) between consecutive odd numbers.$	$a_{1} = 23 a_{n} = 75 step = 2$
$Next, we count how many terms appear in this evenly spaced list. By the "Counting Evenly Spaced Integers" rule, we subtract the endpoints, divide by the step, and add one because the list is inclusive.$	$n = \frac{75 - 23}{2} + 1 = \frac{52}{2} + 1 = 26 + 1 = 27$	Theory & Tactics Method Card EVEN1-B Click to view full details
$We then find the average of the sequence, which for an evenly spaced list is the mean of the first and last terms.$	$average = \frac{23 + 75}{2} = \frac{98}{2} = 49$
$Finally, we apply the "Sum of Evenly Spaced Integers" rule: the sum equals the average multiplied by the number of terms.$	$sum = 49 \times 27 = (49 \times 20) + (49 \times 7) = 980 + 343 = 1323$	Theory & Tactics Method Card EVEN1-D Click to view full details

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B