Q.Select the correct answer from the given alternatives. If for a G.P. $\dfrac{t_6}{t_3}=\dfrac{1458}{54}$ then $r=$? A) 3 B) 2 C) 1 D) $-1$

General Term of a Geometric Progression

The Intuition: What a G.P. feels like

Imagine you start with a single grain of rice on the first square of a chessboard. On the second square, you put 2 grains. On the third, 4 grains. On the fourth, 8 grains. Each time, you multiply the previous number by 2.

That is a geometric progression. The defining rule is simple: you get the next term by multiplying the current term by a fixed number (called the common ratio, usually denoted $r$). In the rice example, $r = 2$.

So if the first term is $a$, the sequence looks like:

Notice the pattern: the exponent on $r$ is always one less than the position of the term. The 1st term has $r^0$, the 2nd has $r^1$, the 3rd has $r^2$, and so on.

The Precise Statement

For a geometric progression with first term $a$ and common ratio $r$ ($r \neq 0$), the $n$th term (also called the general term) is given by:

where $n$ is a positive integer ($n = 1, 2, 3, \dots$).

Why This Formula Works

Let's verify it for the first few terms:

• $n = 1$: $T_1 = a r^{0} = a$ (correct)
• $n = 2$: $T_2 = a r^{1} = ar$ (correct)
• $n = 3$: $T_3 = a r^{2} = ar^2$ (correct)

The formula is simply a compact way of saying: "start with $a$, then multiply by $r$ exactly $(n-1)$ times."

A Common Mistake to Avoid

What If $r$ Is Negative or a Fraction?

The formula works for any real $r$ (except $r=0$, which makes the sequence trivial). If $r = -2$, the terms alternate sign: $a, -2a, 4a, -8a, \dots$. If $r = \frac{1}{3}$, the terms shrink: $a, \frac{a}{3}, \frac{a}{9}, \frac{a}{27}, \dots$. The same $T_n = a r^{n-1}$ handles both cases perfectly.

A Quick Example

Find the 10th term of the G.P.: $3, 6, 12, 24, \dots$

Here $a = 3$ and $r = 6/3 = 2$. Using $T_n = a r^{n-1}$:

The 10th term is 1536.

The Big Picture

The general term is the DNA of the progression — it tells you the value at any position without having to list all the previous terms. Once you know $a$ and $r$, you know the entire infinite sequence. This idea (describing a pattern with a single formula) is one of the most powerful in mathematics.