Understanding the nth Term of an Arithmetic Progression (AP): Formula and Applications

The nth term of an Arithmetic Progression (AP) helps us find any term in a sequence without listing all the preceding terms. This is particularly useful when dealing with large sequences or finding terms positioned far into the sequence.

Formula for the nth Term

The formula to find the nth term of an AP is:

$\boldsymbol{a_n = a + (n - 1)d}$

Where:

$\boldsymbol{a_n}$ = the nth term of the AP,
$\boldsymbol{a}$ = the first term of the AP,
$\boldsymbol{d}$ = the common difference between consecutive terms,
$\boldsymbol{n}$ = the position of the term to find.

This formula provides a direct way to calculate the value of a specific term in an AP without listing all terms up to that point.

General Form of an AP

We know that an Arithmetic Progression (AP) is a sequence of numbers where each term differs from the previous one by a constant value, called the common difference ( $\boldsymbol{d}$ ).
The general form of an AP is:
$\boldsymbol{a, , a + d, , a + 2d, , a + 3d, , \dots, a + (n - 1)d}$ .

Here:

$\boldsymbol{a}$ = the first term of the AP,
$\boldsymbol{d}$ = the common difference,
$\boldsymbol{n}$ = the position of the term in the sequence.

Step-by-Step Derivation

Start with the sequence structure:
The first term of the sequence is $\boldsymbol{a}$ .
The second term is $\boldsymbol{a + d}$ (adding the common difference once).
The third term is $\boldsymbol{a + 2d}$ (adding the common difference twice).
By continuing this pattern, the $\boldsymbol{n}$ th term will have the common difference added $\boldsymbol{(n - 1)d}$ times.
Write the general expression for the nth term:
To obtain the $\boldsymbol{n}$ th term, start with the first term ( $\boldsymbol{a}$ ) and add $\boldsymbol{(n - 1)d}$ multiples of the common difference ( $\boldsymbol{d}$ ):
$\boldsymbol{a_n = a + (n - 1)d}$ .

Verification

Let’s verify the formula with an example.

Example: Verify the formula for the AP $\boldsymbol{3, 7, 11, 15, \dots}$ .

First term ( $\boldsymbol{a}$ ): $\boldsymbol{3}$ ,
Common difference ( $\boldsymbol{d}$ ): $\boldsymbol{7 - 3 = 4}$ ,
Find the 4th term using the formula:
$\boldsymbol{a_4 = a + (n - 1)d = 3 + (4 - 1)(4)}$ .
$\boldsymbol{a_4 = 3 + 3(4) = 3 + 12 = 15}$ .

The 4th term is $\boldsymbol{15}$ , which matches the sequence.

Examples with Step-by-Step Solutions

Example 1: Find the 15th term of the AP $\boldsymbol{3, 7, 11, 15, \dots}$ .

Solution:

Identify the first term and the common difference:
$\boldsymbol{a = 3, , d = 7 - 3 = 4}$ .
Use the formula for the nth term:
$\boldsymbol{a_n = a + (n - 1)d}$ .
Substitute the values of $\boldsymbol{a}$ , $\boldsymbol{d}$ , and $\boldsymbol{n}$ :
$\boldsymbol{a_{15} = 3 + (15 - 1)(4)}$ .
Simplify the expression:
$\boldsymbol{a_{15} = 3 + 14(4) = 3 + 56 = 59}$ .

Answer: The 15th term of the AP is $\boldsymbol{a_{15} = 59}$ .

Example 2: Find the 20th term of the AP $\boldsymbol{-5, -1, 3, 7, \dots}$ .

Solution:

Identify the first term and the common difference:
$\boldsymbol{a = -5, , d = -1 - (-5) = -1 + 5 = 4}$ .
Use the formula for the nth term:
$\boldsymbol{a_n = a + (n - 1)d}$ .
Substitute the values of $\boldsymbol{a}$ , $\boldsymbol{d}$ , and $\boldsymbol{n}$ :
$\boldsymbol{a_{20} = -5 + (20 - 1)(4)}$ .
Simplify the expression:
$\boldsymbol{a_{20} = -5 + 19(4) = -5 + 76 = 71}$ .

Answer: The 20th term of the AP is $\boldsymbol{a_{20} = 71}$ .

Example 3: The 12th term of an AP is 35, and the common difference is 3. Find the first term.

Solution:

Use the formula for the nth term:
$\boldsymbol{a_n = a + (n - 1)d}$ .
Substitute the known values:
$\boldsymbol{35 = a + (12 - 1)(3)}$ .
Simplify the equation:
$\boldsymbol{35 = a + 11(3)}$
$\boldsymbol{35 = a + 33}$ .
Solve for $\boldsymbol{a}$ :
$\boldsymbol{a = 35 - 33 = 2}$ .

Answer: The first term of the AP is $\boldsymbol{a = 2}$ .

Example 4: Which term of the AP $\boldsymbol{2, 5, 8, 11, \dots}$ is 47?

Solution:

Use the formula for the nth term:
$\boldsymbol{a_n = a + (n - 1)d}$ .
Substitute the known values:
$\boldsymbol{47 = 2 + (n - 1)(3)}$ .
Simplify the equation:
$\boldsymbol{47 = 2 + 3(n - 1)}$
$\boldsymbol{47 - 2 = 3(n - 1)}$
$\boldsymbol{45 = 3(n - 1)}$ .
Solve for $\boldsymbol{n}$ :
$\displaystyle\boldsymbol{n - 1 = \frac{45}{3} = 15}$
$\boldsymbol{n = 15 + 1 = 16}$ .

Answer: The 47th term is the $\boldsymbol{16}$ th term of the AP.

Applications of the nth Term Formula

Predicting Future Events:
In real-life scenarios, the nth term formula is used to predict outcomes based on regular patterns, such as financial growth or population changes.
Mathematical Modeling:
It is used to model patterns in nature, such as the arrangement of leaves or the growth of sequences.
Engineering:
Engineers use APs to design systems with evenly spaced components or sequences.

Practice Questions

Find the 25th term of the AP $\boldsymbol{10, 15, 20, 25, \dots}$ .
Answer: $\boldsymbol{a_{25} = 130}$ .
Which term of the AP $\boldsymbol{7, 14, 21, \dots}$ is 77?
Answer: $\boldsymbol{a_{11} = 77}$ .
The 8th term of an AP is 24, and the common difference is 2. Find the first term.
Answer: $\boldsymbol{a = 10}$ .
If the 5th term of an AP is 0 and the 15th term is 20, find the common difference and the first term.
Answer: $\boldsymbol{d = 2, a = -8}$ .

FAQs

How is the nth term derived?admin2024-12-18T13:52:37+05:30

nth Term of an AP

Formula for the nth Term

General Form of an AP

Step-by-Step Derivation

Verification

Example: Verify the formula for the AP .

Examples with Step-by-Step Solutions

Example 1: Find the 15th term of the AP .

Example 2: Find the 20th term of the AP .

Example 3: The 12th term of an AP is 35, and the common difference is 3. Find the first term.

Example 4: Which term of the AP is 47?

Applications of the nth Term Formula

Practice Questions

FAQs

Related Topics

Related Posts

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Example: Verify the formula for the AP $\boldsymbol{3, 7, 11, 15, \dots}$ .

Example 1: Find the 15th term of the AP $\boldsymbol{3, 7, 11, 15, \dots}$ .

Example 2: Find the 20th term of the AP $\boldsymbol{-5, -1, 3, 7, \dots}$ .

Example 4: Which term of the AP $\boldsymbol{2, 5, 8, 11, \dots}$ is 47?