Q.If the first term of a GP is 5 and common ratio is $(-5)$, then which term is 3125?

What is an Arithmetic Progression?

Imagine you're climbing stairs. Each step takes you up by the same height — say 15 cm. After one step you're at 15 cm, after two steps at 30 cm, after three at 45 cm, and so on. The sequence of heights is: 15, 30, 45, 60, 75, …

That's the core idea of an Arithmetic Progression (AP) : a list of numbers where you keep adding the same fixed number to get the next term.

The Intuition

An AP is the simplest pattern in numbers: constant growth (or constant shrinkage). Every step forward changes the value by the same amount.

• If you start at 2 and add 3 each time: 2, 5, 8, 11, 14, …
• If you start at 100 and subtract 10 each time: 100, 90, 80, 70, …

That fixed number you add (or subtract) is called the common difference, usually denoted by $d$. It can be positive, negative, or even zero (a constant sequence like 7, 7, 7, …).

The Precise Definition

The first term is usually denoted by $a$ (or $a_1$). So the AP looks like:

The General Term (nth term)

If you want the 10th term, you don't write out all 10 numbers. Notice the pattern:

• 1st term: $a$
• 2nd term: $a + d$
• 3rd term: $a + 2d$
• 4th term: $a + 3d$

The coefficient of $d$ is always one less than the term number. So the $n$th term is:

This is the single most important formula in AP. It lets you jump directly to any term.

Sum of the First n Terms

Sometimes you need the total of the first $n$ terms. There's a beautiful trick: pair the first term with the last, the second with the second-last, and so on. Each pair sums to the same number.

If the first term is $a$ and the last term is $l = a + (n-1)d$, then:

Or, substituting $l$:

A Quick Example

Problem: The 5th term of an AP is 17 and the 10th term is 32. Find the first term and common difference.

Solution:

We have:

• $a_5 = a + 4d = 17$
• $a_{10} = a + 9d = 32$

Subtract the first equation from the second:

Then $a + 4(3) = 17 \implies a + 12 = 17 \implies a = 5$.

So the AP is: 5, 8, 11, 14, 17, 20, …

Why It Matters

Arithmetic progressions appear everywhere: simple interest calculations, salary increments, depreciation of assets, and even in patterns of natural phenomena. Once you see the constant-step pattern, you've spotted an AP — and you know exactly how to describe it with just two numbers: $a$ and $d$.