Final answer:
By solving the equation 3x-7=27-4x, we find that x=4.857. Given the value of x, we can calculate the common difference of the arithmetic progression, which is 11.571, and then use the sum formula to find the sum of the first 10 terms, which is 607.835.
Step-by-step explanation:
To find the value of x that satisfies the equation 3x−7=27−4x, we first need to simplify the equation by collecting like terms. Adding 4x to both sides gives 7x−7=27, and then adding 7 to both sides gives 7x=34. Dividing both sides by 7 yields x=34/7, which simplifies to x=4.857. Using this x value, we can proceed to find the sum of the first 10 terms of the arithmetic progression 2x−1, 5x−4, 8x−7,….
First, we determine the common difference d by subtracting consecutive terms: (5x−4)-(2x−1) = 3x−3. Plugging x=4.857 into this yields d=11.571. The first term a1 is 2x−1, which is 2(4.857)−1 = 8.714. To find the sum of the first 10 terms S10, we use the formula S10 = n/2(2a1 + (n−1)d), where n is the number of terms. Substituting the known values, S10 = 10/2(2(8.714) + (10−1)(11.571)), which simplifies to S10 = 5(17.428 + 9(11.571)), and finally to S10 = 5(17.428 + 104.139) = 5(121.567) = 607.835.