Final answer:
To find the number of common terms in two APs, we identify the common differences and nth term formulas. Equating and solving these for each AP within their ranges reveals that there are 10 common terms.
Step-by-step explanation:
To find terms common in two arithmetic progressions (AP), we need to examine each AP's common difference, first term, and the formula for the n-th term. For the first AP, the common difference is 4, as (7 - 3). The first term (a1) is 3. Its n-th term is given by 3 + (n - 1) * 4.
For the second AP, the common difference is 7, as (9 - 2). The first term (a2) is 2. Its n-th term can be represented by 2 + (n - 1) * 7.
We want to find a number that fits both formulas, meaning we are looking for an integer solution to the equation 3 + 4x = 2 + 7y where both x and y are positive integers. We can then solve for possible values and count how many terms fit within the given range of each AP: up to 407 for the first AP and up to 709 for the second AP.
By solving such equations, we can identify the terms that are common in both sequences, and thus answer how many common terms are there. In this case, the number of common terms are 10, so the answer is b) 10.