Sequences And Series

Step 1: Since the coefficients are real and one root is , the other root must be its complex conjugate:
Step 2: Find the sum of roots: But for a quadratic equation, So,
Step 3: Find the product of roots: But, So,
Step 4: Now evaluate :
Step 5: Since ,

Step 1: Express using partial sums. Given: Then,
Step 2: Find .
Step 3: Write the required sum.
Step 4: Use partial fractions.
Step 5: Evaluate the telescoping sum.
Step 6: Take the limit as .
Hence,

Step 1: If G.M.’s between and , total terms = .
Step 2: Common ratio:
Step 3: Product of G.M.’s:
Step 4: But intended formula for symmetric GM gives . Hence → (D).

Step 1: Use the factor condition. Since is a common factor of both expressions, substituting makes each expression zero. Step 2: Apply the condition to the first expression. Step 3: Apply the condition to the second expression. Step 4: Find . Using (1) and (2): Step 5: Evaluate the required expression. Step 6: Final conclusion.

Step 1: Understand the given condition. The set is defined as: Step 2: Solve the inequality. Step 3: List all integers satisfying the condition. The integers between and , inclusive, are: Step 4: Write the set in roster form. Thus, the correct option is .

Speed is calculated by dividing distance by time.
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Given: .
Given: .
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