Final answer:
To find the term of the arithmetic progression that is 2015, we first find the common difference by using the given ratio and the formula for arithmetic progression. Next, we find the first term by using the given square difference and the formula for arithmetic progression. Finally, we use the formula for arithmetic progression and the value of the 1st term to find the value of n, which represents the term of the series.
Step-by-step explanation:
To solve this problem, we can use the formulas for arithmetic progression and quadratic equations. Let's start by finding the common difference (d) of the arithmetic progression. We are given that the ratio of the difference between the 4th and 8th terms to the 15th term is 4/15. Using the formula for arithmetic progression, we have (8th term - 4th term) / 15th term = 4/15. Simplifying this equation, we get (4d) / (4 + 4d) = 4/15. Solving for d, we find d = 5/6.
Next, we can use the given information that the square difference of the 4th and 1st term is 225. Using the formula for arithmetic progression, we have (4th term - 1st term)^2 = 225. Substituting the value of d, we get (4(5/6) - 1st term)^2 = 225. Simplifying this equation, we get (20/6 - 1st term)^2 = 225. Solving for the 1st term, we find the 1st term to be 5/6.
Now, we can find the value of n, which represents the term of the series. Using the formula for arithmetic progression, we have a_n = a_1 + (n-1)d, where a_n represents the nth term, a_1 is the 1st term, and d is the common difference. Substituting the values we found earlier, we have a_n = 5/6 + (n-1)(5/6). Setting a_n equal to 2015 and solving for n, we find n = 2015. Therefore, the term of the series that is 2015 is option c) 2015.