Introduction and Arithmetic Progressions

Patterns All Around Us

Look at these lists: the savings of ₹100, ₹150, ₹200, ₹250, … each month; or seats 20, 22, 24, 26, … in successive rows of a theatre. Notice something? Each number is obtained by adding the same fixed amount to the previous one.

Such a list is called an Arithmetic Progression (AP).

Definition: An arithmetic progression is a list of numbers in which each term (after the first) is obtained by adding a fixed number $d$ to the preceding term. The fixed number $d$ is called the common difference.

Think of it this way: an AP grows (or shrinks) in equal steps. The step size is $d$.

[Board Important] The key test for an AP: the difference between any term and its previous term is always the same value $d$.

First Term, Common Difference, and General Form

An AP is fully described by two numbers:

• the first term, written $a$ (or $a_1$);
• the common difference $d = a_2 - a_1 = a_3 - a_2 = \dots$ (any term minus the one before it).

The general form of an AP is:

$$a, \ a + d, \ a + 2d, \ a + 3d, \ \dots$$

Example

For the AP $3, 7, 11, 15, \dots$: $a = 3$ and $d = 7 - 3 = 4$. So the terms are $3, 3+4, 3+2(4), \dots$

Key Point: To find $d$, subtract any term from the term that follows it: $d = a_{n+1} - a_n$. Always subtract in this order (later minus earlier).

[Board Important] $d$ can be positive (increasing AP), negative (decreasing AP), or zero (all terms equal). Compute it from at least two consecutive differences to be sure it's constant.

Checking Whether a List is an AP

To test if a list forms an AP, compute the differences between consecutive terms. If all these differences are equal, it's an AP; otherwise it isn't.

Worked outline

• $2, 5, 8, 11, \dots$: differences $3, 3, 3, \dots$ — equal → AP ($d = 3$).
• $1, 4, 9, 16, \dots$: differences $3, 5, 7, \dots$ — not equal → not an AP.

Key Point: Don't check just one difference — verify that several consecutive differences are the same before concluding it is an AP.

[Board Important] Squares ($1, 4, 9, \dots$) and other non-linear patterns are not APs. An AP must increase/decrease by a constant amount each step.

Finite and Infinite APs

An AP can have a limited number of terms or continue forever.

• A finite AP has a fixed number of terms and a last term (denoted $l$ or $a_n$). Example: $2, 4, 6, \dots, 100$.
• An infinite AP continues without end. Example: $5, 10, 15, 20, \dots$

We usually write three dots '…' to indicate continuation.

Key Point: A finite AP has a last term $l$; an infinite AP does not. The number of terms in a finite AP is denoted $n$.

Building an AP from $a$ and $d$

Given $a = 6$ and $d = -2$, the AP is $6, 4, 2, 0, -2, \dots$ (a decreasing AP).

[Board Important] When a problem gives $a$ and $d$, you can immediately write out as many terms as needed by repeatedly adding $d$.

Solved Examples

Example 1: Find $a$ and $d$

For the AP $5, 8, 11, 14, \dots$, write the first term and common difference.

Solution:

1. First term $a = 5$.
2. $d = 8 - 5 = 3$ (check: $11 - 8 = 3$, $14 - 11 = 3$). ✓

Final Answer: $a = 5$, $d = 3$.

Takeaway: $d$ = any term minus the previous term.

Example 2: Decreasing AP

Find $a$ and $d$ for $10, 7, 4, 1, \dots$

Solution:

1. $a = 10$.
2. $d = 7 - 10 = -3$.

Final Answer: $a = 10$, $d = -3$.

Takeaway: A negative $d$ means a decreasing AP.

Example 3: Is it an AP?

Does $2, 4, 8, 16, \dots$ form an AP?

Solution:

1. Differences: $4 - 2 = 2$, $8 - 4 = 4$, $16 - 8 = 8$.
2. The differences ($2, 4, 8$) are not equal.

Final Answer: No, it is not an AP.

Takeaway: Unequal consecutive differences ⇒ not an AP (this is a GP).

Example 4: Confirm an AP

Does $-3, -1, 1, 3, \dots$ form an AP? If so, give $d$.

Solution:

1. Differences: $-1 - (-3) = 2$, $1 - (-1) = 2$, $3 - 1 = 2$.
2. All equal to 2.

Final Answer: Yes, AP with $d = 2$.

Takeaway: Be careful subtracting negatives: $-1 - (-3) = 2$.

Example 5: Write terms from $a$ and $d$

Write the first four terms of the AP with $a = 7$ and $d = -2$.

Solution:

1. $a_1 = 7$; $a_2 = 7 + (-2) = 5$; $a_3 = 5 - 2 = 3$; $a_4 = 3 - 2 = 1$.

Final Answer: $7, 5, 3, 1$.

Takeaway: Repeatedly add $d$ to generate the terms.

Example 6: AP with fractions

Find $d$ for the AP $\dfrac{1}{2}, 1, \dfrac{3}{2}, 2, \dots$

Solution:

1. $d = 1 - \dfrac{1}{2} = \dfrac{1}{2}$ (check: $\dfrac{3}{2} - 1 = \dfrac{1}{2}$). ✓

Final Answer: $d = \dfrac{1}{2}$.

Takeaway: Fractional common differences are perfectly valid.

Example 7: Find a missing term

The first three terms of an AP are $x, 2x + 1, 5x - 2$. Find $x$.

Solution:

1. In an AP, $a_2 - a_1 = a_3 - a_2$.
2. $(2x + 1) - x = (5x - 2) - (2x + 1) \Rightarrow x + 1 = 3x - 3$.
3. $4 = 2x \Rightarrow x = 2$.

Final Answer: $x = 2$.

Takeaway: Equal consecutive differences give an equation for the unknown.

Example 8: Form an AP from a situation

A man saves ₹500 in the first month and increases his saving by ₹50 each month. Write the AP of his monthly savings.

Solution:

1. First term $a = 500$, common difference $d = 50$.
2. AP: $500, 550, 600, 650, \dots$

Final Answer: $500, 550, 600, 650, \dots$ (in ₹).

Takeaway: 'Increases by a fixed amount each time' signals an AP.

Example 9: General form check

Is the list given by $a_n = 3n + 2$ an AP? Find $a$ and $d$.

Solution:

1. $a_1 = 5$, $a_2 = 8$, $a_3 = 11$.
2. Differences are all 3, so it is an AP with $a = 5$, $d = 3$.

Final Answer: AP with $a = 5$, $d = 3$.

Takeaway: If $a_n$ is linear in $n$ (like $3n + 2$), the list is always an AP; $d$ is the coefficient of $n$.

Example 10: Find $d$ from two terms' difference

In an AP, the difference between the 4th and 1st terms is 9. Find $d$.

Solution:

1. $a_4 - a_1 = (a + 3d) - a = 3d$.
2. $3d = 9 \Rightarrow d = 3$.

Final Answer: $d = 3$.

Takeaway: $a_4 - a_1 = 3d$; in general $a_m - a_n = (m - n)d$.