Arithmetic Progression

Sum of all positive divisors There is only one divisor of which is greater than So, sum of all divisors less than

We are given that are in arithmetic progression (AP) and are in geometric progression (GP). Step 1: Express the terms in terms of and Since are in AP, we know that: Next, since are in GP, we know that: Substitute and : Simplifying the equation: This simplifies to . Step 2: Find the ratio Substitute into the expressions for and : Thus, the ratio is: Thus, the correct answer is .

Let the first term of the AP be and the common difference be . The formula for the nth term of an AP is given by: The sum of the nth term and the mth term is: Substitute for and : Simplify to: Now solve for in terms of : Now, to find the sum of the first 33 terms of the AP, we use the formula for the sum of the first n terms: For : Substitute : Simplifying this expression:

For three numbers to form both an Arithmetic Progression ((A)P) and a Geometric Progression (G.P), they must satisfy the conditions for both sequences. 1. For (A)P: The condition for to be in Arithmetic Progression is that the middle term should be the average of the other two terms: 2. For G.P: The condition for to be in Geometric Progression is that the square of the middle term should be the product of the other two terms: Now, substituting into the G.P condition: Simplifying the equation: Expanding both sides: Rearranging terms: This is the equation of a perfect square: Therefore, . Since and we already have , we conclude that: Thus, the answer is .