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Sum Of First N Terms Of An AP

Derive and apply the sum formula for an Arithmetic Progression.

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When the famous mathematician Carl Friedrich Gauss was just 10 years old, his teacher asked the class to add the numbers from 1 to 100. Gauss did it almost instantly! He wrote $S = 1 + 2 + \dots + 100$ , reversed it to $S = 100 + 99 + \dots + 1$ , and added them together to get $2 S = 100 \times 101$ , concluding the sum was 5050.

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Sum of First n Terms of an AP Visual

A visual representation of Gauss's reversal method makes the derivation stick.

bold editorial infographic, clean data-forward design, high-contrast color blocks, elegant typography hierarchy, geometr…

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Deriving the Sum Formula

Generalizing Gauss's method to any AP.

Writing the Sequence

To find the sum of the first $n$ terms of an AP, let's denote the sum as $S$ .

The general form of an AP is $a, a+d, a+2d, \dots$
The $n$ th term of this AP is $a + (n-1)d$ .

We can write the sum $S$ as:
$S = a + (a+d) + (a+2d) + \dots + [a + (n-1)d]$

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Sum Formulas

The two main forms of the sum formula.

The Standard Formula

S_{n} = \frac{n}{2} [2 a + (n - 1) d]

When to use: You know the first term ( $a$ ), the common difference ( $d$ ), and the number of terms ( $n$ ), but you do NOT know the last term.

The Short Formula (First + Last)

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

When to use: You know the first term ( $a$ ) and the last term ( $l$ or $a_{n}$ ), along with the total number of terms ( $n$ ). It saves time by skipping the common difference calculation.

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Using the Main Sum Formula

Example 11: Sum of 22 terms.

Problem Statement

Example 11: Find the sum of the first 22 terms of the AP: $8, 3, -2, \dots$

Given Data:
First term, $a = 8$
Common difference, $d = 3 - 8 = -5$
Number of terms, $n = 22$

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Finding a Term Given the Sum

Example 12: Given S_14 = 1050.

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We are given that the sum of the first 14 terms is 1050, so the value of $S_{14}$ is . We also know the first term $a = 10$ .

To find the 20th term, we must first figure out the common difference $d$ . Using the sum formula $S_n = \frac{n}{2}[2a + (n-1)d]$ , we substitute to get $1050 = \frac{14}{2}[2(10) + 13d]$ .

Solving this gives $1050 = 7[20 + 13d]$ , which simplifies to $150 = 20 + 13d$ . Thus, $d$ equals .

Now, to find the 20th term, we switch to the general term formula $a_{20} = a + 19d$ . Substituting our known values gives $a_{20} = 10 + 19(10)$ , which evaluates to .

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Formula Selection

Test understanding of which formula to apply.

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An engineer is designing a stadium seating layout. The layout acts as an AP where the first row has 5 seats, the last row has 45 seats, and the total capacity is 400 seats. What is the most efficient first formula to use to find the total number of rows ( $n$ )?

← previousApplications Of The Nth Term next →Advanced Sum Applications

Deriving the Sum Formula

Generalizing Gauss's method to any AP.

Writing the Sequence

To find the sum of the first $n$ terms of an AP, let's denote the sum as $S$ .

The general form of an AP is $a, a+d, a+2d, \dots$
The $n$ th term of this AP is $a + (n-1)d$ .

We can write the sum $S$ as:
$S = a + (a+d) + (a+2d) + \dots + [a + (n-1)d]$

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Using the Main Sum Formula

Example 11: Sum of 22 terms.

Problem Statement

Example 11: Find the sum of the first 22 terms of the AP: $8, 3, -2, \dots$

Given Data:
First term, $a = 8$
Common difference, $d = 3 - 8 = -5$
Number of terms, $n = 22$

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Finding a Term Given the Sum

Example 12: Given S_14 = 1050.

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We are given that the sum of the first 14 terms is 1050, so the value of $S_{14}$ is . We also know the first term $a = 10$ .

To find the 20th term, we must first figure out the common difference $d$ . Using the sum formula $S_n = \frac{n}{2}[2a + (n-1)d]$ , we substitute to get $1050 = \frac{14}{2}[2(10) + 13d]$ .

Solving this gives $1050 = 7[20 + 13d]$ , which simplifies to $150 = 20 + 13d$ . Thus, $d$ equals .

Now, to find the 20th term, we switch to the general term formula $a_{20} = a + 19d$ . Substituting our known values gives $a_{20} = 10 + 19(10)$ , which evaluates to .

Formula Selection

Test understanding of which formula to apply.

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