The third term of a G.P. is 4, the product of the first five terms is ................ .

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Concept understanding — Geometric Progression

Geometric Progression: The Idea of Repeated Multiplication

Imagine you're folding a piece of paper in half. Start with thickness 1 unit. After one fold, thickness becomes 2. After two folds, thickness becomes 4. After three folds, thickness becomes 8. The sequence of thicknesses is:

1, 2, 4, 8, 16, ...

Notice the pattern: each term is obtained by multiplying the previous term by the same number (here, 2). That's the core intuition behind a geometric progression — you keep multiplying by a fixed number, step after step.

This is different from an arithmetic progression, where you keep adding a fixed number. Here, the growth is multiplicative, not additive. That's why geometric progressions grow (or shrink) much faster.

Precise Definition

A Geometric Progression (GP) is a sequence of numbers where the ratio of any term to its preceding term is constant. This constant is called the common ratio, denoted by $r$ .

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

a, \ ar, \ ar^2, \ ar^3, \ ar^4, \ \dots

Note

The common ratio $r$ can be any real number — positive, negative, or even a fraction. If $r$ is negative, the terms alternate in sign. If $0 < r < 1$ , the terms get smaller and smaller.

The $n$ -th Term

To find any term directly without listing all previous ones, use the formula:

T_n = a \cdot r^{\,n-1}

where $T_n$ is the $n$ -th term, $a$ is the first term, $r$ is the common ratio, and $n$ is the term number (starting from 1).

Example: For the paper-folding sequence, $a = 1$ , $r = 2$ . The 5th term is $1 \cdot 2^{5-1} = 2^4 = 16$ , which matches our list.

Sum of $n$ Terms

There are two cases, depending on whether $r = 1$ or not.

n

When $r = 1$ , every term is just $a$ , so the sum is simply $n \times a$ .

Why the formula works (intuition):

Let $S = a + ar + ar^2 + \dots + ar^{n-1}$ . Multiply both sides by $r$ : $rS = ar + ar^2 + \dots + ar^n$ . Subtract the first from the second: $rS - S = ar^n - a$ , so $S(r-1) = a(r^n - 1)$ , giving the formula above.

Sum of an Infinite GP

If the common ratio $r$ lies strictly between $-1$ and $1$ (i.e., $|r| < 1$ ), the terms get smaller and smaller, and the sum of all terms approaches a finite value:

S_\infty = \frac{a}{1 - r}, \quad \text{for } |r| < 1

Watch out

If $|r| \geq 1$ , the infinite sum does not exist (it diverges to infinity or oscillates without settling). Never apply the infinite sum formula when $|r| \geq 1$ .

Example: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ has $a=1$ , $r=\frac12$ , so $S_\infty = \frac{1}{1 - 1/2} = 2$ . This matches the intuition that repeatedly halving a unit length eventually fills exactly 2 units.

Quick Reference Table

Property	Formula	Condition
Common ratio	$r = \dfrac{T_{n+1}}{T_n}$	Always
$n$ -th term	$T_n = a r^{n-1}$	Always

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In a geometric progression, the product of the first five terms equals the fifth power of the middle (third) term. Since the third term is $4$ , the product is $4^5 = 1024$ .

A geometric progression (GP) is a sequence where each term after the first is obtained by multiplying the previous term by a fixed constant called the common ratio ( $r$ ). The key insight here is symmetry: when you multiply five consecutive terms of a GP, the middle term appears with a special power.

Let the first term be $a$ and the common ratio be $r$ . Then the terms are:

First term: $a$
Second term: $ar$
Third term: $ar^2$
Fourth term: $ar^3$
Fifth term: $ar^4$

We are told the third term is $4$ , so:

ar^2 = 4

Now, the product of the first five terms is:

P = a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4

Count the powers of $a$ and $r$ separately.

There are five $a$ 's multiplied together, so $a^5$ .

For $r$ , the exponents add: $0 + 1 + 2 + 3 + 4 = 10$ , so $r^{10}$ .

Thus:

P = a^5 \cdot r^{10}

Rewrite the product in terms of the third term.

Notice that $a^5 r^{10} = (a r^2)^5$ . Why? Because $(ar^2)^5 = a^5 (r^2)^5 = a^5 r^{10}$ .

This is the elegant symmetry: the product of five terms in a GP is the fifth power of the middle term.
$2n+1$
Substitute the given value.

Since $ar^2 = 4$ , we get: …

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Q.The third term of a G.P. is 444, the product of the first five terms is ................ .

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Q.The third term of a G.P. is $4$ , the product of the first five terms is ................ .

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