Sequences And Series

Step 1: Evaluate the expression for :
Consider the expression .
Substituting , Factorizing , we get Thus, the possible common divisors are and .

Step 2: Evaluate the expression for :
Now substitute , Check divisibility:
For 11: Using the alternating sum rule, So, is divisible by .
For 19: Hence, is not divisible by .
Therefore, the only common divisor is .

Step 3: Nature of the divisor:
The number is both odd and prime.
Thus, is an odd prime number.
Hence, the correct option is (C).

The given inequation is for all .
We can rearrange the inequality to solve for :
.
This inequality must hold for all natural numbers (i.e., ).
Therefore, must be less than or equal to the minimum value of the expression .
Let . We need to find the minimum value of for .
Let's test the first few values of :
For : .
For : .
For : .
The values of are increasing as increases. This indicates the minimum value of occurs at the smallest value of , which is .
The minimum value of the expression is .
The condition becomes , which is .
The question asks for the greatest integer value of that satisfies this condition.
The greatest integer less than or equal to is .

Step 1: Understanding the Concept
The given series is a special type of series where each term can be expressed as the difference of two consecutive terms of another sequence. This is known as a telescoping series. The sum can be found by writing out the terms and observing that the intermediate terms cancel each other out.
Step 2: Key Formula or Approach
First, find the general n-th term, , of the series. The denominators consist of products of numbers in an arithmetic progression. We can use the method of partial fractions to decompose into the form .
Step 3: Detailed Explanation
The sequence of the first factors in the denominators is 2, 7, 12, 17, ... This is an Arithmetic Progression (AP) with first term and common difference . The n-th term is .
The sequence of the second factors is 7, 12, 17, 22, ... This is also an AP with first term and common difference . The n-th term is .
So, the general n-th term of the series is:
We use partial fractions to decompose . Setting , we get . Setting , we get . So, the n-th term can be written as:
We need to find the sum of the first 10 terms, .
Let's write out the terms:
For n=1:
For n=2:
For n=3:
...
For n=10:
When we add all these terms, the intermediate terms cancel out.
Step 4: Final Answer
The sum of the series to 10 terms is .