Arithmetic Progression

Let the first term be and the common difference be . The 5th term is given by: The 12th term is given by: Subtract (i) from (ii): Substitute in equation (i): Now, the sum of first 20 terms:

The correct option is (D):

Step 1: Recall the formula for the -th term of an AP.

The -th term of an AP is given by:

where is the first term and is the common difference.

Step 2: Identify known terms and find the common difference .

The first term . The third term is , and the fourth term is .

Using the formula for the -th term:

Substitute and :

Step 3: Find the second term ( ).

The second term is given by:

Final Answer: The missing term is .

Correct answer: 27

Explanation:
In an Arithmetic Progression (A.P.), the middle term is the average of the first and third terms.
So,

Hence, the value of is 27.

Correct answer: 25

Explanation:
The sequence is: 781, 806, 831, 781, 806, 831, ...
This is a repeating pattern and not a standard Arithmetic Progression.
However, if we consider only the first three terms: 781, 806, 831
Then: So, the common difference is 25

Correct answer: 2b

Explanation:
If are in Arithmetic Progression, then: Multiplying both sides by 2:

Hence,

In an arithmetic progression (A.P.), the general term is given by:

Given that the 25th term is 70 more than the 15th term:

Using the general formula:

Canceling from both sides:

Thus, the common difference is: .

To find the value of n such that the numbers n – 3, 4n – 2, and 5n + 1 are in arithmetic progression, we need to use the property of an arithmetic progression, which states that twice the middle term is equal to the sum of the first and third terms.

This can be expressed as:

2(4n - 2) = (n - 3) + (5n + 1)

Expanding both sides, we get:

8n - 4 = n - 3 + 5n + 1

Simplifying further,

8n - 4 = 6n - 2

To solve for n, subtract 6n from both sides:

8n - 6n - 4 = -2

2n - 4 = -2

Adding 4 to both sides gives:

2n = 2

Dividing both sides by 2 to solve for n, we get:

n = 1

Thus, the value of n is 1.

1. The given arithmetic progression (A.P.) is:

5, 1, −3, −7, . . .

2. The general form of an A.P. is:

The nth term of an A.P. is given by:

Where is the first term, is the common difference, and is the term number.

3. Finding the common difference:

The common difference is the difference between any two consecutive terms:

4. Finding the 10th term:

Substitute , , and into the formula: