Arithmetic Progression

We are given that the 10^th and 12^th terms of an arithmetic progression (A.P.) are 15 and 21, respectively. We are asked to find the common difference of the A.P.

The formula for the term of an A.P. is:

where is the term, is the first term, and is the common difference.

We are given:

Using the formula for the term, we can write two equations:

Now, we have the system of equations:

Subtract the first equation from the second to eliminate :

The common difference of the A.P. is 3.

We are given that in an arithmetic progression (A.P.), the difference between the last term and the first term is 632, and the common difference is 4. We are asked to find the number of terms in the A.P.

The formula for the term of an A.P. is:

where is the first term, is the common difference, and is the -\) term.

We are given that the common difference , and the difference between the last and first terms is 632. The last term can be written as:

Using the formula for the term:

Simplifying:

Substitute into the equation:

Solving for :

Therefore,

The number of terms in the A.P. is 159.

We are given the arithmetic progression (A.P.) with the first term and the common difference , and we are asked to find the value of for which .

The -th term of an A.P. is given by the formula:

Substituting the given values and , we get:

We are given that

Now, solve for :

The value of is 101.

We are given that are in arithmetic progression (A.P.), with We need to find the value of

We know the sum of the first terms of an A.P. is given by the formula:

where is the first term, is the common difference, and is the number of terms.

We are also given that the sum of the terms is 210, so:

Substitute into the equation:

We also know that , and

Now substitute this value of into the earlier equation:

The value of is 10.

We are given the sum of the first terms of the series:

The -th term of the series is given by:

Step 1: Find and

From the given formula for , we have: Now, substitute into the formula for to find : Expanding this:

Step 2: Find

Now, subtract from to find : Simplify the expression: Thus, the -th term is:

The correct option is (A) :

In an Arithmetic Progression (A.P.), the terms follow the rule: Where is the nth term, is the first term, and is the common difference. We are given that the last three terms are: We can find the common difference by subtracting any two consecutive terms: Now, using the formula for the nth term, we can express as: Substitute into the equation: Solving for :

The correct option is (D) :

We are given an arithmetic series:

This is an arithmetic progression (AP) where:

• First term
• Common difference

We need to find the sum of the first 24 terms of this series. The formula for the sum of the first terms of an arithmetic progression is:

Substituting the values , , and into the formula, we get:

Thus, the sum of the first 24 terms is .

The correct option is (D) :

Let the sides of the right-angled triangle be in arithmetic progression: where . Since it's a right-angled triangle, by the Pythagorean theorem: Expanding: Simplifying: Since , we have . The area of the triangle is: Substitute : Thus: The sides are: The longest side (hypotenuse) is:

Let the sequence be where and . Given: Expanding: From equation (1): . Substitute into (2): Since the sequence is increasing with natural numbers, . Then: Final answer: