Arithmetic Progression

If are in A.P., then we need to find the value of the determinant:

Since the terms are in A.P., we can write , where d is the common difference.

So, , , , , , , , .

Substituting these into the determinant:

We can perform row operations without changing the value of the determinant. Apply and :

Since the second and third rows are proportional (i.e., ), the determinant is 0.

Thus, (C) is zero.

The sum of n terms of an A.P. is given by .

We need to find the common difference, denoted by 'd'.

We know that the sum of the first n terms of an A.P. can also be represented as:

where 'a' is the first term and 'd' is the common difference.

We are given .

Let's find and :

Since represents the sum of the first term, it is equal to the first term itself:

represents the sum of the first two terms, so:

We know , so:

Therefore, the common difference is 2.

To find the sum of the first 51 terms of an Arithmetic Progression (A.P.) given that its middle term is 300, we can follow these steps:

1. Since there are 51 terms in the A.P., the middle term is the 26th term. In an A.P. with an odd number of terms, the middle term is given by the formula for the general nth term of an A.P., which is: where is the first term, is the common difference, and is the term number.
2. For the 26th term: (Equation 1)
3. To find the sum of the first 51 terms, we use the sum formula for an A.P.: where is the sum of the first terms.
4. Substitute into the formula:
5. From Equation 1, we have:
6. Now substitute back into the sum formula:

Thus, the sum of the first 51 terms of the A.P. is 15300.

Therefore, the correct answer is 15300.

The sum of the first terms of an arithmetic progression (A.P.) is given by the formula: where:
is the first term,
is the common difference,
is the number of terms.
In this case, the number of terms is 51, and the middle term is given as 300. Since the middle term of an A.P. is the -th term, we have: Simplifying, we get: Now, using the formula for the sum of the first 51 terms: Substituting into the equation: Therefore, the sum of the first 51 terms is .