Sequences

The given sum is:

Step 1: Split the Terms of the Numerator
Rewrite the numerator as:

Thus, the sum becomes:

Step 2: Simplify Each Term Using Series Expansions
For the first term:

This is the series expansion of , so:

For the second term:

Thus:

For the third term

Thus:

Step 3: Combine All Terms
Combine all terms:

Simplify:

Combine terms:


Conclusion
The sum is:

To solve for the product of the middle two terms of the given arithmetic progression (AP), follow these steps:

Step 1: Understand the AP Parameters
Given an AP: a₁=8, a₂, a₃, ..., a_n. The sum of the first four terms is 50, and the sum of the last four terms is 170.

Step 2: Find the Common Difference
The first four terms are: a₁=8, a₂, a₃, a₄. The sum is:

8 + (8 + d) + (8 + 2d) + (8 + 3d) = 50

32 + 6d = 50

6d = 18

d = 3

Step 3: Determine n
The last four terms are: a_n-3, a_n-2, a_n-1, a_n, with sum:

(8 + (n-4)d) + (8 + (n-3)d) + (8 + (n-2)d) + (8 + (n-1)d) = 170

4(8) + (4n - 10)d = 170

32 + 12n - 30 = 170

12n = 168

n = 14

Step 4: Calculate Middle Terms and Their Product
With n = 14, the AP terms are: a₁ = 8, a₂=11, ..., a₁₄. Middle terms: a₇, a₈.

a₇ = 8 + 6d = 8 + 18 = 26

a₈ = 8 + 7d = 8 + 21 = 29

The product: 26 × 29 = 754

Step 5: Validate Solution
The product of the middle terms, 754, falls within the specified range (754,754). Thus, this is the correct and validated solution.

Given:

Simplify the expression: Using the formula for :

Now, consider: Summing over from 1 to 10:

Expand and calculate: Substitute and simplify further using telescoping:

After calculation: The final result is:

Final Answer: The value of is 6.

The correct answer is 222.
The logic follows here is :
2³ + 2 = 10
3³ + 3 = 30
4³ + 4 = 68
5³ + 5 = 130
6³ + 6 = 222
7³ + 7 = 350
Therefore , the missing number is 222.