Introducing Arithmetic Progressions

Define an AP, identify its first term and common difference, and construct a sequence from them.

7 lessons

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What is an Arithmetic Progression?

Define AP, terms, general form, and common difference.

An Arithmetic Progression (AP) is a list of numbers in which each term is obtained by adding a fixed number to the preceding term, except the first term.

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Introducing Arithmetic Progressions Visual

A comparison table contrasting APs with non-APs reinforces the definition.

$clean side-by-side comparison, consistent visual treatment for both sides, same scale and style, pastel color coding to distinguish categories, elegant typography, white background, generous whitespace. Left side shows a valid Arithmetic Progression with numbers 2, 4, 6, 8 connected by evenly spaced +2 arrows. Right side shows a list of squares with numbers 1, 4, 9, 16 connected by unevenly growing arrows (+3, +5, +7), clearly marked as NOT an Arithmetic Progression.$

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An AP must have a constant step size (common difference) between every term.

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The Common Difference

Formula for finding d.

d = a_{k + 1} - a_{k}

Subtract the

k

th term from the

(k + 1)

th term

Example

For the sequence: $6, 3, 0, - 3$

$d = a_{2} - a_{1} = 3 - 6 = - 3$

Important Watch-out

Always subtract the preceding term from the succeeding term. Do NOT just subtract the smaller number from the larger number! If the sequence is decreasing, $d$ will be negative.

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Finding a and d

Worked Example 1 from the text.

Problem. For the AP: $\frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}, \dots$ , write the first term $a$ and the common difference $d$ .

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Checking for an AP

Faded Example 2(ii) from the text.

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To check if the sequence $1, -1, -3, -5, \dots$ is an AP, we verify if the difference between consecutive terms is the same every time. First, we calculate $a_2 - a_1$ , which is $-1 - 1 =$ . Next, we calculate $a_3 - a_2$ , which gives $-3 - (-1) =$ . Because the difference is identical every time, this list forms an AP with a common difference of $-2$ . The next two terms are calculated by adding the common difference: $-5 + (-2) =$ . Following that, the term after will be $-7 + (-2) =$ .

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quiz

Spot the AP

Identify which sequence is an AP.

Which of the following lists of numbers forms an Arithmetic Progression (AP)?

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Write the AP

Generate the first four terms of an AP given a and d.

Given

a = -1.25

and

d = -0.25

, generate the first four terms of the AP.

Use the general form of an AP ( $a, a+d, a+2d, a+3d$ ) to calculate the first four terms.

Given Data*

List the known values.

General Form*

State the pattern you will use.

Substitution*

Show how you plug the values in for the 2nd, 3rd, and 4th terms.

Calculation*

Calculate the arithmetic values.

Final Sequence*

List the four consecutive terms separated by commas.

← backNCERT X - Mathematics next →The Nth Term Of An AP

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Finding a and d

Worked Example 1 from the text.

Problem. For the AP: $\frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}, \dots$ , write the first term $a$ and the common difference $d$ .

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reading

Checking for an AP

Faded Example 2(ii) from the text.

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quiz

Spot the AP

Identify which sequence is an AP.

Which of the following lists of numbers forms an Arithmetic Progression (AP)?

feedback

Write the AP

Generate the first four terms of an AP given a and d.

Given

a = -1.25

and

d = -0.25

, generate the first four terms of the AP.

Use the general form of an AP ( $a, a+d, a+2d, a+3d$ ) to calculate the first four terms.

Given Data*

List the known values.

General Form*

State the pattern you will use.

Substitution*

Show how you plug the values in for the 2nd, 3rd, and 4th terms.

Calculation*

Calculate the arithmetic values.

Final Sequence*

List the four consecutive terms separated by commas.