Final answer:
The sum of the first 50 terms of the arithmetic sequence with a common difference of 4, starting with the first term 3, is 5050.
Step-by-step explanation:
To find the sum of the first 50 terms of the arithmetic sequence with the first term 3 and the third term 11, we need to determine the common difference and then use the formula for the sum of an arithmetic series. The common difference (d) can be found by taking the difference between the third and first terms and dividing by the number of steps between them. We get d = (11 - 3) / (3 - 1) = 4.
Using the formula for the sum of the first n terms of an arithmetic sequence Sn = n/2 * (2a + (n - 1)d), where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms, we get:
S₅₀ = 50/2 * (2*3 + (50 - 1)*4)
S₅₀ = 25 * (6 + 49*4)
S₅₀ = 25 * (6 + 196)
S₅₀ = 25 * 202
S₅₀ = 5050
Therefore, the sum of the first 50 terms of the given arithmetic sequence is 5050.