Write the formula for the sum of terms in AP.

To find the sum of the first \( n \) terms in an arithmetic progression (AP), we can use a specific formula that simplifies the calculation. The formula is:

Sum of the First n Terms in an AP

The sum \( S_n \) of the first \( n \) terms of an arithmetic progression can be calculated using the formula:

S_n = \frac{n}{2} \times (2a + (n - 1)d)

In this formula:

• S_n represents the sum of the first \( n \) terms.
• a is the first term of the AP.
• d is the common difference between consecutive terms.
• n is the number of terms to be summed.

Alternative Formulation

There’s another way to express this sum, which can be particularly useful:

S_n = \frac{n}{2} \times (a + l)

Here, \( l \) is the last term of the sequence. This formulation is handy when you know the last term instead of the common difference.

Understanding the Components

Let’s break down the formula a bit further:

• The term \( 2a \) accounts for the first term being counted twice when you add the first and last terms together.
• The term \( (n - 1)d \) adjusts for the number of times the common difference is added to the first term to reach the last term.

Example for Clarity

Suppose you have an AP where the first term \( a = 3 \) and the common difference \( d = 2 \). If you want to find the sum of the first 5 terms:

• The first term \( a = 3 \).
• The common difference \( d = 2 \).
• The number of terms \( n = 5 \).

Using the first formula:

S_n = \frac{5}{2} \times (2 \times 3 + (5 - 1) \times 2)

S_n = \frac{5}{2} \times (6 + 8) = \frac{5}{2} \times 14 = 5 \times 7 = 35

So, the sum of the first 5 terms of this AP is 35.

Visualizing the AP

To visualize, the terms of this AP would be:

• 3 (first term)
• 5 (3 + 2)
• 7 (5 + 2)
• 9 (7 + 2)
• 11 (9 + 2)

When you add these terms together (3 + 5 + 7 + 9 + 11), you indeed get 35, confirming our formula works perfectly!

In summary, the formula for the sum of the first \( n \) terms in an arithmetic progression is a powerful tool that allows you to quickly calculate the total without having to add each term individually. Understanding how to apply it will greatly enhance your ability to work with sequences and series in mathematics.