Sequence And Series

given

Given series, The th terms of the series is Now,

The correct answer is (V) : .

The correct answer is (D) : n(n - 1)2^n-2.

Step 1: Understanding the Concept:
Inserting numbers between two given numbers and to form a Geometric Progression (G.P.) implies creating a sequence of terms where the first term is , the last term is , and the sequence has a constant common ratio. The inserted numbers are called geometric means.

Step 2: Key Formula or Approach:
Let the newly formed G.P. sequence be . We are given and . The general formula for the -th term of a G.P. is . We can use the 4th term to solve for the common ratio . Once is found, the inserted numbers are and .

Step 3: Detailed Explanation:
Let the two numbers inserted be and . The resulting sequence is: . This sequence is a Geometric Progression. The first term is . Since there are 4 terms in total, the fourth term is . Let be the common ratio of this G.P. Using the formula for the -th term, , we have: Substitute the known values into the equation: Now, solve for : To simplify, multiply the numerator and denominator by to rationalize the denominator: We can express as a cube to easily find : So, we have: Taking the real cube root of both sides gives the common ratio: Now we calculate the inserted numbers and , which are the second and third terms of the G.P.: First inserted number, . Second inserted number, . Therefore, the inserted numbers in order are and .

Step 4: Final Answer:
The inserted numbers are .

Step 1: Write the general term
Let
Step 2: Transform the term
Rewrite as :
Step 3: Use factorial property
So,
Step 4: Form the series

Step 5: Expand the series

Step 6: Apply telescoping
All intermediate terms cancel:
Step 7: Simplify
Final Answer: