Arithmetic Progression

Given are in arithmetic progression, so: Also given: Use sine rule: So: Now use identity: Now use more identities or check ratio from these values: if we take , then , and , then: This contradicts given options. Try , then , From that, using sine values, you'll find

Given that are in arithmetic progression, we can use the property of arithmetic progression. For any three terms in arithmetic progression, the middle term is the average of the other two. Therefore, we have the relation: This implies that the ratio of the radii is . Thus, the correct answer is option (4) .

Let the first term be and common difference be . Sum of first 5 terms: Multiply both sides by 2: Sum of next 5 terms (terms 6 to 10) is given as 235: Sum of first 10 terms: Sum of next 5 terms is: Substitute and : Divide both sides by 5: Subtract equation (1) from equation (2):

Let the three numbers in arithmetic progression be , where is the first term and is the common difference. Since all numbers must lie in the set , we have the constraint: because the largest term must be at most 100. For each fixed , the number of valid choices for is: The common difference can take integer values from 1 up to: Therefore, total number of 3-term arithmetic progressions is:

The sum of the first terms of an arithmetic progression is: Given , , : Thus, the common difference is:

The sequence has terms. Arithmetic mean (AM) is given by: Sum , so