2.6. Arithmetic Progression (A.P.)

● A sequence where the difference between consecutive terms is constant.

An Arithmetic Progression (A.P.) is a special kind of number sequence where each term after the first is created by adding a fixed number (called the common difference) to the previous term. For example, in the sequence 2, 5, 8, 11, the common difference is 3, since 5 - 2 = 3, 8 - 5 = 3, and 11 - 8 = 3.

Imagine you save money every month. If you save an initial amount of $10 and add$5 each month, your savings create a pattern: $10,$15, $20,$25, and so on. This pattern is an A.P. where the difference between each month's savings is always $5.

● nth term: an=a+(n−1)d a_n = a + (n - 1)d

To find the nth term of an Arithmetic Progression, you can use the formula: an = a + (n - 1)d, where 'a' is the first term, 'd' is the common difference, and 'n' is the term number you are calculating. This formula helps determine what the value of any term in the sequence is, depending on its position.

If you have a series of numbers in an A.P., like 3, 7, 11, ..., and you want to find the 5th term, you can apply the formula. Here, a = 3, d = 4 (because 7 - 3 = 4), and n = 5. So, substituting into the formula gives you 3 + (5 - 1) × 4 = 3 + 16 = 19. Thus, the 5th term is 19.

● Sum of n terms: Sn=n2[2a+(n−1)d] S_n = \frac{n}{2}[2a + (n - 1)d]

The sum of the first n terms of an Arithmetic Progression can be calculated using the formula: Sn = n/2 [2a + (n - 1)d]. Here, 'S_n' represents the sum of the first n terms, 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms you wish to sum. This formula simplifies the process of finding the total when adding many terms together.

If you want to calculate how much money you've saved after the first 5 months of saving $10 initially and adding$5 every month, you can determine the sum of the series 10, 15, 20, 25, 30. Here, a = 10, d = 5, and n = 5. Using the formula, you can find that amount: S5 = 5/2 [2*10 + (5 - 1)*5] = 5/2 [20 + 20] = 5/2 * 40 = 100. So, you would have saved a total of $100 after 5 months.

✦ Example: Find the 10th term of the A.P.: 5, 8, 11, … Solution: a = 5, d = 3 a10=5+(10−1)×3=5+27=32 a_{10} = 5 + (10 - 1) × 3 = 5 + 27 = 32

In this example, we need to find the 10th term of the sequence: 5, 8, 11, ... , where the first term 'a' is 5 and the common difference 'd' is 3. Using the nth term formula, we substitute n = 10 to calculate a10. Thus, a_{10} = 5 + (10 - 1) × 3 = 5 + 27 = 32. The 10th term in this A.P. is therefore 32.

Think of a game where you gain points in equal increments. If you start with 5 points and gain 3 points each round, at the end of the 10th round, you can calculate your total points easily using the A.P. formula. After 10 rounds, you'll have accumulated 32 points.

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