Question

Use the formula for the sum of the first n integers to evaluate the sum given below, then write it in closed form.

(a) 8+9+10+11+ . . . . +500

(b) 8+9+10+11+ . . . +k

k^2+k-72 / 2

Answer 1

The sum of integers from 8 to 500 using the sum formula is 125476. For an arbitrary final term k, the sum in closed form is (k^2 + 2k - 48) / 2, which differs slightly from the provided expression.

Step-by-step explanation:

The question asks to evaluate the sum of a sequence of integers and express it in a closed form. For part (a), we are looking at the sum of integers from 8 to 500, and for part (b), from 8 to an arbitrary integer k. We can use the formula for the sum of the first n integers which states that the sum of the first n terms of an arithmetic sequence is (n/2)*(first term + last term). We apply this formula to solve both parts of the question.

To find the sum from 8 to 500, we calculate the total number of terms, n, by subtracting 7 (one less than 8) from 500 and then adding 1. Thus, n is 500 - 7 + 1 = 494. Using the sum formula, we get (494/2)*(8+500) = 247*(508) = 125476.

The sum from 8 to k has n = k - 7 terms. Again, using the sum formula, we get ((k - 7 + 1)/2)*(8 + k) = (k - 6)/2*(k + 8) = (k^2 + 2k - 48) / 2. The provided expression k^2 + k - 72 is similar, with a slight variation in the constant term, and dividing by 2 gives the sum in closed form.

Answer 2

The sum of the integers from 8 to 500 is calculated using the sum of the first n integers formula and equals 125,028. For the sum from 8 to k, where k is an arbitrary number, the expression k(k + 1) / 2 - 36 represents the closed form of the sum.

Step-by-step explanation:

The student is asking for the sum of a sequence of integers using a formula. Specifically, there are two parts:

To evaluate and express in closed form the sum of the sequence from 8 to 500.

To evaluate and write in closed form the sum of the sequence from 8 to an arbitrary number k, where the sum is given by the expression (k^2 + k - 72) / 2.

To solve part (a), we use the formula for the sum of the first n integers, which is (n/2) * (first term + last term). Here, n is the total number of terms.

To find n, we calculate n = 500 - 8 + 1 = 493. So the sum is (493/2) * (8 + 500) = 125,028.

For part (b), since we already have the sum in a semi-closed form, we simplify the expression to k(k + 1) / 2 - 36, which represents the sum of the integers from 8 to k minus the sum of the integers from 1 to 7 (which is 28).