Arithmetic And Geometric Progressions

This is an infinite series with a common ratio of . We can write the series as: This is a geometric series. By setting up the sum of the series and solving for , we find its value.
Final Answer: .

We are given the terms of an A.P. and need to find the value of .
Step 1: Use the given values and to solve for the common difference .
Step 2: Use the general formula for the term of an A.P. to express in terms of .
Step 3: Use the sum formula for an A.P. to find .
Step 4: Multiply the result by to compute the final value.

Final Conclusion: The value of is 126, which is Option 3.

Given the term of an arithmetic progression (A.P.) , where is the first term and is the common difference.

We are given:

We know: .

Also, .

From , we solve for and .

First, rewrite the sum equation:

Solving simultaneously with previous equations:

Eliminate by subtracting the two:

Calculate:

Substitute in :

To compute :

Thus, :

Upon solving with given and values: yields:

This is an infinite series with a common ratio of . We can write the series as: This is a geometric series. By setting up the sum of the series and solving for , we find its value.
Final Answer: .

To solve this problem, we will first determine the expression for the general term of the given series and use it to find the sum . Then, we will relate the information given about the arithmetic progression (A.P.) to compute the absolute difference between the 20th and 15th terms.

1. Let's express the general formula for the sequence . These terms suggest they follow the pattern of the form: .
2. We need to compute the sum : .
3. The term can be decomposed using partial fraction: . This is a telescoping series, and it simplifies to: .
4. The series reduces to: .
5. Now, the sum of the first six terms of the A.P. given is: . Substituting the value of : .
6. Since the first term of the A.P. is and the common difference is , the sum of the first six terms: . Simplify: .
7. Compute the absolute difference between the 20th and 15th terms of the A.P.: .
   • 20th term:
   • 15th term:

Thus, the absolute difference between the 20th and 15th terms of the A.P. is 25.

Step 1: We are given a sum involving factorial terms, and we need to compute the limit as . We can simplify the expression by examining the asymptotic behavior of the sum.
Step 2: For large , the term grows very quickly compared to the polynomial terms in the numerator. Thus, the terms of the sum decrease rapidly as increases.
Step 3: We recognize that the sum is dominated by the first few terms, and we compute the value of the infinite sum by summing the first few terms and taking the limit. The value of the sum as is . Thus, the correct answer is (1).