Q.If $a, b$ and $c$ are in A.P. as well as in G.P., then which of the following is true?

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 $l$:

$S_n = \frac{n}{2} \left[ 2a + (n-1)d \right]$ …