Final answer:
The value of t5/t9 in the arithmetic progression, where the second and fourth terms have a ratio of 3:7, is also 3:7. The property of an arithmetic progression ensures that the ratio remains the same for any two terms equidistant from each other.
Step-by-step explanation:
To solve this problem, we need to understand the properties of an arithmetic progression (AP). An AP is defined by its first term, a, and its common difference, d. The nth term, denoted tn, of an arithmetic progression is given by the formula tn = a + (n - 1) * d. Therefore, the second term, t2, is a + d, and the fourth term, t4, is a + 3d.
Given that t2 / t4 = 3/7, we can set up the equation (a + d) / (a + 3d) = 3/7. Solving for a in terms of d, we find that a equals -3d or a and d have a specific ratio that satisfies the given condition. However, this ratio does not impact the ratio of any two terms with equal intervals in an AP, because the common difference d will be multiplied by the same number for each term in the ratio.
Considering the fifth term, t5, and the ninth term, t9, the interval between them is four terms, just like the interval between t2 and t4. Hence, t5 / t9 = (a + 4d) / (a + 8d) will simplify to the same ratio as t2 / t4 = 3/7 because the a terms cancel out. Thus, the value of t5 / t9 is also 3/7.