Q.The first term of an A.P. is $a$, and the sum of the first $p$ terms is zero, show that the sum of its next $q$ terms is $\dfrac{-a(p+q)q}{p-1}$. [Hint: Required sum $= S_{p+q} - S_p$]

Arithmetic Progression: The Pattern of Equal Steps

Imagine you're climbing a staircase where every step has the exact same height. If the first step takes you to 3 feet, and each step after that adds exactly 2 feet, your heights would be: 3, 5, 7, 9, 11, ... That's an arithmetic progression — a sequence where you move forward by adding the same number every time.

The Core Idea

An Arithmetic Progression (AP) is a list of numbers where the difference between any two consecutive terms is constant. This constant is called the common difference, usually denoted by $d$.

If the first term is $a$, then the sequence looks like:

The pattern is simple: you start at $a$, then keep adding $d$ to get the next term.

The General Term (nth term)

What if you want the 100th term without writing all 100 numbers? There's a formula.

The first term is $a$ (think of it as $a + 0 \cdot d$).

The second term is $a + d$ (that's $a + 1 \cdot d$).

The third term is $a + 2d$.

Notice the pattern: the term number minus 1 tells you how many times $d$ has been added.

So the nth term (also called the general term) is:

Example: For the AP 3, 5, 7, 9, ... we have $a = 3$, $d = 2$.

The 10th term: $T_{10} = 3 + (10-1) \cdot 2 = 3 + 18 = 21$.

Why "Arithmetic"?

The name comes from an old property: in an AP, every term (except the first and last) is the arithmetic mean of its neighbours. For three consecutive terms $x, y, z$ in an AP:

Check: in 3, 5, 7, we have $5 = \frac{3+7}{2} = 5$. This works for any three consecutive terms.

Sum of the First n Terms

Sometimes you need the total of the first $n$ terms. There's a clever trick.

Write the sum forwards: $S_n = a + (a+d) + (a+2d) + \dots + [a+(n-1)d]$

Write it backwards: $S_n = [a+(n-1)d] + [a+(n-2)d] + \dots + a$

Add them term by term. Each pair adds to $2a + (n-1)d$, and there are $n$ such pairs. So:

Therefore:

There's another useful form. Since the last term $l = a + (n-1)d$, we can write:

This is beautiful: the sum of an AP is just the number of terms times the average of the first and last term.

Example: Sum of first 10 terms of 3, 5, 7, ...

$S_{10} = \frac{10}{2}[2\cdot3 + (10-1)\cdot2] = 5[6 + 18] = 5 \times 24 = 120$

Quick Reference

Common Mistakes to Avoid

• Confusing $n$ with the term value. $n$ is the position (1st, 2nd, 3rd...), not the number itself.
• Forgetting the $(n-1)$ in the nth term. Many students write $a + nd$ by mistake. The first term has zero $d$'s added, so it's $a + (1-1)d = a$.
• Using the wrong $n$ in the sum formula. If you want the sum of the first 20 terms, $n = 20$, not 21.

A Real-World Feel

APs show up everywhere: monthly rent increasing by a fixed amount each year, the number of seats in each row of an auditorium (if each row has 2 more seats than the previous), or even the simple act of counting by 5s: 5, 10, 15, 20, ... That's an AP with $a=5$, $d=5$.

Once you see the pattern of equal steps, you'll spot arithmetic progressions all around you.

Arithmetic Progression is one of the most exam-heavy topics in the NCERT Class 11 Mathematics chapter on Sequences and Series, matching frequent searches for "arithmetic progression nth term and sum formula" or "AP important questions class 11 maths". Its equal-step pattern also shows up regularly in JEE Main and state CET numerical-ability sections, often disguised as real-world word problems like EMIs or seating arrangements.