Introduction to Arithmetic Progressions: Concepts, Formulas, and Examples

Q: What is an Arithmetic Progression (AP)?

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Arithmetic Progressions (AP) are an integral part of mathematics, widely used in various fields such as finance, physics, and computer science. Understanding APs helps in solving problems related to sequences, patterns, and series. This guide covers the basics of AP, its formulas, properties, and real-life applications with examples.

What is an Arithmetic Progression (AP)?

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference ( $\boldsymbol{d}$ ).

General Form of an AP:

$\boldsymbol{a, \, a + d, \, a + 2d, \, a + 3d, \, \dots}$

Where:

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

For example:

$\boldsymbol{2, 4, 6, 8, 10, \dots}$ (here, $\boldsymbol{a = 2}$ and $\boldsymbol{d = 2}$ ),
$\boldsymbol{10, 7, 4, 1, -2, \dots}$ (here, $\boldsymbol{a = 10}$ and $\boldsymbol{d = -3}$ ).

Terms of an Arithmetic Progression

The $\boldsymbol{n}$ th term of an AP, denoted by $\boldsymbol{a_n}$ , is given by the formula:

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

Where:

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

Sum of an Arithmetic Progression

The sum of the first $\boldsymbol{n}$ terms of an AP is given by the formula:

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

Alternatively, if the first term ( $\boldsymbol{a}$ ) and the last term ( $\boldsymbol{l}$ ) are known:

$\displaystyle\boldsymbol{S_n = \frac{n}{2} (a + l)}$

Where:

$\boldsymbol{S_n}$ = sum of the first $\boldsymbol{n}$ terms,
$\boldsymbol{a}$ = first term,
$\boldsymbol{l}$ = last term,
$\boldsymbol{n}$ = number of terms.

Properties of Arithmetic Progression

The difference between any two consecutive terms is always constant.
The sequence can increase (if $\boldsymbol{d > 0}$ ), decrease (if $\boldsymbol{d < 0}$ ), or remain constant (if $\boldsymbol{d = 0}$ ).
The average of the first and last terms is equal to the average of any pair of terms equidistant from the beginning and end.

Examples

Example 1: Find the 10th term of an AP where $\boldsymbol{a = 5}$ and $\boldsymbol{d = 3}$ .

Solution:
Using the formula $\boldsymbol{a_n = a + (n - 1)d}$ :

$\boldsymbol{a_{10} = 5 + (10 - 1)(3) = 5 + 27 = 32}$ .

Answer: The 10th term is $\boldsymbol{32}$ .

Example 2: Find the sum of the first 15 terms of an AP where $\boldsymbol{a = 7}$ and $\boldsymbol{d = 2}$ .

Solution:
Using the formula $\displaystyle\boldsymbol{S_n = \frac{n}{2} \big[2a + (n - 1)d\big]}$ :

$\displaystyle\boldsymbol{S_{15} = \frac{15}{2} \big[2(7) + (15 - 1)(2)\big]}.$ . $\displaystyle\boldsymbol{S_{15} = \frac{15}{2} \big[14 + 28\big] = \frac{15}{2} (42) = 315}.$ .

Answer: The sum of the first 15 terms is $\boldsymbol{315}$ .

Example 3: The 5th term of an AP is 20, and the 15th term is 50. Find the first term and the common difference.

Solution:
Using the formula $\boldsymbol{a_n = a + (n - 1)d}$ :
For the 5th term:

$\boldsymbol{a_5 = a + 4d = 20}$ .

For the 15th term:

$\boldsymbol{a_{15} = a + 14d = 50}$ .

Subtracting the first equation from the second:

$\boldsymbol{(a + 14d) - (a + 4d) = 50 - 20}$ . $\boldsymbol{10d = 30 \quad \Rightarrow \quad d = 3}$ .

Substituting $\boldsymbol{d = 3}$ into $\boldsymbol{a + 4d = 20}$ :

$\boldsymbol{a + 4(3) = 20 \quad \Rightarrow \quad a = 8}$ .

Answer: The first term is $\boldsymbol{8}$ , and the common difference is $\boldsymbol{3}$ .

Applications of Arithmetic Progressions

Finance: Calculating interest payments in loans or deposits with fixed increments.
Physics: Analyzing uniform motion where equal distances are covered in equal intervals of time.
Engineering: Designing structures with evenly spaced components.
Daily Life: Arranging seats, distributing items, or determining schedules with constant intervals.

Practice Questions

Find the 12th term of an AP where $\boldsymbol{a = 3}$ and $\boldsymbol{d = 5}$ .
Answer: $\boldsymbol{a_{12} = 58}$ .
The sum of the first 10 terms of an AP is $\boldsymbol{155}$ , and the common difference is $\boldsymbol{2}$ . Find the first term.
Answer: $\boldsymbol{a = 6}$ .
The 4th term of an AP is $\boldsymbol{10}$ , and the 8th term is $\boldsymbol{22}$ . Find the common difference and the first term.
Answer: $\boldsymbol{d = 3, a = 1}$ .