is the sum of positive integers between (inclusive) and (inclusive) that are not multiples of and not multiples .
Step-by-step explanation:
For an arithmetic series with:
as the first term,
as the last term, and
as the common difference,
there would be terms, where as the sum would be .
Positive integers between (inclusive) and (inclusive) include:
.
The common difference of this arithmetic series is . There would be terms. The sum of these integers would thus be:
.
Similarly, positive integers between (inclusive) and (inclusive) that are multiples of include:
.
The common difference of this arithmetic series is . There would be terms. The sum of these integers would thus be:
Positive integers between (inclusive) and (inclusive) that are multiples of include:
.
The common difference of this arithmetic series is . There would be terms. The sum of these integers would thus be:
Positive integers between (inclusive) and (inclusive) that are multiples of (integers that are both multiples of and multiples of ) include:
.
The common difference of this arithmetic series is . There would be terms. The sum of these integers would thus be:
.
The requested sum will be equal to:
the sum of all integers from to ,
minus the sum of all integer multiples of between and , and the sum integer multiples of between and ,
plus the sum of all integer multiples of between and - these numbers were subtracted twice in the previous step and should be added back to the sum once.
Given the radius, circumference can be solved by the equation, C = 2πr. The circumference of the circle above is C = 2π(8 in) = 16<span>π in. To solve for the length of the segment joining the arc is the circumference times the ratio of central angle and 360 degrees.
Length of the segment = (16</span>π in)(60/360) = 8/3 <span>π in
Thus, the length of the segment is approximately 8.36 in. </span>