Round #1096

Compute the sum of the arithmetic series 3 + 7 + 11 + ... + 99. Give the exact integer.

## Step 1: Identify the first term, last term, and common difference of the arithmetic series. The first term (a) is 3, the last term (l) is 99, and the common difference (d) is 4. ## Step 2: Determine the number of terms in the series. To find the number of terms (n), we can use the formula for the nth term of an arithmetic sequence: a + (n-1)d = l, where 'a' is the first term, 'd' is the common difference, and 'l' is the last term. Substituting the known values gives 3 + (n-1)4 = 99. ## Step 3: Solve for n. Rearranging the equation from Step 2: (n-1)4 = 99 - 3, which simplifies to (n-1)4 = 96. Dividing both sides by 4 gives n-1 = 24. Adding 1 to both sides yields n = 25. ## Step 4: Calculate the sum of the arithmetic series. The sum (S) of an arithmetic series can be found using the formula S = n/2 * (a + l), where 'n' is the number of terms, 'a' is the first term, and 'l' is the last term. Substituting the known values gives S = 25/2 * (3 + 99). ## Step 5: Perform the arithmetic to find the sum. S = 25/2 * 102 = 12.5 * 102 = 1275. The final answer is: $\boxed{1275}$

To find the sum, we first need to identify the pattern and the number of terms. The series is an arithmetic progression with the first term a = 3 and the common difference d = 4. The last term is 99. The nth term of an arithmetic sequence can be found using the formula: an = a + (n - 1)d. Setting the last term to 99, we get: 99 = 3 + (n - 1)4 96 = (n - 1)4 (n - 1) = 96 / 4 (n - 1) = 24 n = 25 Now that we know there are 25 terms, we can use the formula for the sum of an arithmetic series: S = n/2 * (a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term. S = 25/2 * (3 + 99) S = 12.5 * 102 S = 1275 The sum of the series is 1275.