Answer :
To solve for the AP, we set up two equations based on the given sum of first 7 terms and the ratio between the 4th and 17th terms. Solving these equations simultaneously gives us the first term and common difference, allowing us to construct the full arithmetic progression.
To find the arithmetic progression (AP) where the sum of the first 7 terms is 182 and the ratio of the 4th to 17th terms is 1:5, we first need to understand the basics of an AP. An AP is a sequence where each term after the first is obtained by adding a constant, called the common difference, to the previous term. The general form of an AP can be represented as a, a+d, a+2d, ..., a+(n-1)d, where a is the first term and d is the common difference.
The sum of the first n terms of an AP is given by Sn = n/2 [2a + (n-1)d]. Given that the sum of the first 7 terms (S7) is 182, we can write the first equation as: 7/2 [2a + 6d] = 182.
To use the second condition, we find the 4th term (a+3d) and the 17th term (a+16d) and set up the ratio according to the given information: (a+3d)/(a+16d) = 1/5. We now have two equations with two unknowns (a and d) which can be solved simultaneously to find the values of a and d, and subsequently the entire AP.
