Sum Of First N Terms Of An Ap

The given problem requires evaluating the limit as of the expression:

This requires simplifying both the numerator and the denominator separately before taking the limit as .

1. Consider the numerator:

• Expand it:

• This is a polynomial expression, and for the limit as , we focus on the leading terms:

1. Consider now the denominator:

• Simplify using known formulas:
  • Cubes Sum:

• Squares Sum:

• Leading term of the denominator focusing on

1. For limits, compare the leading terms from numerator and denominator:

Numerator:

Denominator:

1. Now, calculate the limit:

as

Hence, the correct answer is:

.

We are given a series with the first term and a recurrence relation for . We are also given the formula for the sum of the first terms, , and we need to find the value of .

Concept Used:

1. Solving Linear Recurrence Relations: The given relation is a linear first-order non-homogeneous recurrence relation. A common method to solve relations of the form is to divide the entire equation by an appropriate factor (here, ) to transform it into a telescoping sum.

2. Sum of a Geometric Progression (GP): The sum of the first terms of a GP with first term and common ratio is given by:

Step-by-Step Solution:

Step 1: Manipulate the given recurrence relation to find a general form for the term .

The relation is . To simplify this, we divide both sides by :

Step 2: Let's define a new sequence . The relation becomes , which can be written as .

This is a telescoping sum. We can write in terms of :

Step 3: Calculate and find the explicit formula for .

We know , so .

The summation is a geometric series . The first term is , the common ratio is , and there are terms.

Substituting this back into the expression for :

Step 4: Find the general formula for .

Since , we have .

Step 5: Calculate the sum of the first terms, .

Both sums are geometric progressions. For the first sum, . For the second, .

Step 6: Substitute these sums back into the expression for .

Bringing to a common denominator:

Step 7: Equate our derived with the given formula for .

We can cancel from both sides. Also, notice that the term on the left can be factored:

Substituting this back into the equation:

Since the term is non-zero for , we can cancel it from both sides.

Final Computation & Result:

Step 8: Solve the remaining equation for .

Rearranging the terms gives a quadratic equation:

This is a perfect square trinomial:

Solving for , we get:

Therefore, the value of is 6.