Sum of First n Terms of an AP: Formula, Steps, and Examples

The sum of the first $\boldsymbol$ terms of an AP is a widely used formula in mathematics to find the total value of terms in a sequence without manually adding each term. This formula is applicable in solving problems across finance, physics, and engineering.

Formula for the Sum of First n Terms

The sum of the first $\boldsymbol$ terms of an Arithmetic Progression (AP) is denoted by $\boldsymbol$ . There are two primary formulas:

If the first term ( $\boldsymbol$ ) and the common difference ( $\boldsymbol$ ) are known:

$\boldsymbol{S_n = \frac \big[2a + (n - 1)d\big]}$

If the first term ( $\boldsymbol$ ) and the last term ( $\boldsymbol$ ) are known:

$\boldsymbol{S_n = \frac (a + l)}$

Here:

$\boldsymbol$ = the sum of the first $\boldsymbol$ terms,
$\boldsymbol$ = the first term of the AP,
$\boldsymbol$ = the common difference between consecutive terms,
$\boldsymbol$ = the total number of terms,
$\boldsymbol$ = the last term of the AP.

Derivation of the Formula

Using the Sequence Structure

Consider an AP: $\boldsymbol$ .
The sum of the first $\boldsymbol$ terms can be expressed as:
$\boldsymbol$ .

Writing this sum in reverse order:
$\boldsymbol$ .

Add these two equations:
$\boldsymbol$ .

Divide both sides by $\boldsymbol$ :
$\displaystyle\boldsymbol{S_n = \frac \big[2a + (n - 1)d\big]}$ .

Examples with Step-by-Step Solutions

Example 1: Find the sum of the first 15 terms of the AP $\boldsymbol$ .

Solution:

Identify the first term and the common difference:
$\boldsymbol$ .
Use the formula for the sum of the first $\boldsymbol$ terms:
$\displaystyle\boldsymbol{S_n = \frac \big[2a + (n - 1)d\big]}$ .
Substitute the values of $\boldsymbol$ , $\boldsymbol$ , and $\boldsymbol$ :
$\displaystyle\boldsymbol{S_{15} = \frac{15} \big[2(3) + (15 - 1)(4)\big]}$ .
Simplify the expression:
$\displaystyle\boldsymbol{S_{15} = \frac{15} \big[6 + 56\big]}$
$\displaystyle\boldsymbol{S_{15} = \frac{15} \cdot 62 = 465}$ .

Answer: The sum of the first 15 terms is $\boldsymbol$ .

Example 2: The sum of the first 10 terms of an AP is $\boldsymbol$ , and the first term is $\boldsymbol$ . Find the common difference.

Solution:

Use the formula for the sum of the first $\boldsymbol$ terms:
$\displaystyle\boldsymbol{S_n = \frac \big[2a + (n - 1)d\big]}$ .
Substitute the known values:
$\displaystyle\boldsymbol{220 = \frac \big[2(12) + (10 - 1)d\big]}$ .
Simplify the equation:
$\boldsymbol$
$\boldsymbol$ .
Solve for $\boldsymbol$ :
$\boldsymbol$
$\boldsymbol$
$\displaystyle\boldsymbol$
$\displaystyle\boldsymbol$ .

Answer: The common difference is $\displaystyle\boldsymbol$ .

Example 3: Find the sum of all multiples of $\boldsymbol$ between $\boldsymbol$ and $\boldsymbol$ .

Solution:

Form the AP:
The multiples of $\boldsymbol$ between $\boldsymbol$ and $\boldsymbol$ are:
$\boldsymbol$ .
Here, $\boldsymbol$ , $\boldsymbol$ , and $\boldsymbol$ .
Find the number of terms ( $\boldsymbol$ ):
Use the formula for the nth term:
$\boldsymbol$ .
Substituting $\boldsymbol$ , $\boldsymbol$ , and $\boldsymbol$ :
$\boldsymbol$
$\boldsymbol$
$\displaystyle\boldsymbol{n - 1 = \frac{96}}$
$\boldsymbol$ .
Use the formula for the sum of the first $\boldsymbol$ terms:
$\displaystyle\boldsymbol{S_n = \frac (a + l)}$ .
Substituting $\boldsymbol$ , $\boldsymbol$ , and $\boldsymbol$ :
$\displaystyle\boldsymbol{S_{33} = \frac{33} (3 + 99)}$
$\displaystyle\boldsymbol{S_{33} = \frac{33} \cdot 102}$
$\displaystyle\boldsymbol$ .

Answer: The sum of all multiples of $\boldsymbol$ between $\boldsymbol$ and $\boldsymbol$ is $\boldsymbol$ .

Example 4: Find the sum of the first 22 terms of the AP: $\boldsymbol$ .

Solution:

Identify the values:
$\boldsymbol$ , $\boldsymbol$ , $\boldsymbol$ .
Use the formula for the sum of $\boldsymbol$ terms:
$\displaystyle\boldsymbol{S_n = \frac \big[2a + (n - 1)d\big]}$
Substitute the values:
$\displaystyle\boldsymbol{S = \frac{22} \big[2(8) + (22 - 1)(-5)\big]}$
Simplify the expression:
$\boldsymbol$
$\boldsymbol$
$\boldsymbol$
$\boldsymbol$

Answer: $\boldsymbol$

Example 5: If the sum of the first 14 terms of an AP is $\boldsymbol$ , and its first term is $\boldsymbol$ , find the 20th term.

Solution:

Identify the values:
$\boldsymbol$ , $\boldsymbol$ , $\boldsymbol$ .
Use the formula for the sum of $\boldsymbol$ terms:
$\displaystyle\boldsymbol{S_n = \frac \big[2a + (n - 1)d\big]}$
Substitute the values for the 14th term sum:
$\displaystyle\boldsymbol{1050 = \frac{14} \big[2(10) + (14 - 1)d\big]}$
Simplify the equation:
$\boldsymbol$
$\boldsymbol$
$\boldsymbol$
$\boldsymbol$
Find the 20th term using the nth term formula:
$\boldsymbol$
$\boldsymbol$
$\boldsymbol$
$\boldsymbol$

Answer: $\boldsymbol$

Example 6: Find the sum of the first 24 terms of the list of numbers whose nth term is given by $\boldsymbol$ .

Solution:

Expand the sequence:
$\boldsymbol$ , $\boldsymbol$ , $\boldsymbol$ , $\dots$
Identify the values:
$\boldsymbol$ , $\boldsymbol$ , $\boldsymbol$ .
Use the formula for the sum of $\boldsymbol$ terms:
$\displaystyle\boldsymbol{S_n = \frac \big[2a + (n - 1)d\big]}$
Substitute the values:
$\displaystyle\boldsymbol{S_{24} = \frac{24} \big[2(5) + (24 - 1)(2)\big]}$
Simplify the expression:
$\boldsymbol$
$\boldsymbol$
$\boldsymbol$

Answer: $\boldsymbol$

Example 7: A manufacturer produces 600 TV sets in the third year and 700 in the seventh year. Assuming production increases uniformly, find:

The production in the first year,
The production in the 10th year,
The total production in the first 7 years.

Solution:

Since production increases uniformly, it forms an AP. Let the first year production be $\boldsymbol$ and the common difference be $\boldsymbol$ .
From the data:
$\boldsymbol$
$\boldsymbol$
Subtract the first equation from the second:
$\boldsymbol$
Substitute $\boldsymbol$ into $\boldsymbol$ :
$\boldsymbol$
Answer: Production in the first year: $\boldsymbol$ .
Find the production in the 10th year:
$\boldsymbol{a_ = a + (10 - 1)d}$
$\boldsymbol{a_ = 550 + 9(25)}$
$\boldsymbol{a_ = 550 + 225}$
$\boldsymbol{a_ = 775}$
Answer: Production in the 10th year: $\boldsymbol{a_ = 775}$ .
Find the total production in the first 7 years:
$\displaystyle\boldsymbol{S_7 = \frac \big[2a + (7 - 1)d\big]}$
$\displaystyle\boldsymbol{S_7 = \frac \big[2(550) + 6(25)\big]}$
$\displaystyle\boldsymbol{S_7 = \frac \big[1100 + 150\big]}$
$\displaystyle\boldsymbol{S_7 = \frac \cdot 1250}$
$\boldsymbol$

Answer: Total production in the first 7 years: $\boldsymbol$

Practice Problems

Find the sum of the first $\boldsymbol$ terms of the AP $\boldsymbol$ .
Answer: $\boldsymbol{S_ = 1240}$ .
The 8th term of an AP is $\boldsymbol$ , and the sum of the first $\boldsymbol$ terms is $\boldsymbol$ . Find the first term.
Answer: $\boldsymbol$ .
Find the sum of all two-digit numbers divisible by $\boldsymbol$ .
Answer: $\boldsymbol$ .