The third term of a geometric progression is 4. The product of the first five terms is (A) 4³ (B) 4⁵

Q: Wait, why can't we just solve for $a$ and $r$ individually?

That is a brilliant question! In this problem, we are given only one piece of information: the third term is $4$. This gives us the equation $ar^2 = 4$. We have two unknowns, $a$ and $r$, but only one equation. Mathematically, we cannot isolate $a$ or $r$ uniquely. However, the beauty of this problem lies in the fact that we don't need to! We are asked for the product of the terms, and the algebraic structure allows us to find that product directly without knowing the individual values of $a$ and $r$.

Q: How did you know to group them as $(ar^2)^5$?

It comes down to pattern recognition, which is a superpower in JEE Advanced. When we simplified the product to $a^5 r^{10}$, we saw that both exponents ($5$ and $10$) are multiples of $5$. This is a huge hint! It suggests that the entire expression can be written as something raised to the power of $5$. Since we knew $ar^2 = 4$, we looked for a way to create that $ar^2$ inside the parenthesis. By factoring out the $5$, we get $(a^1 r^2)^5$, which is exactly $(ar^2)^5$. Always look for these structural symmetries!

Q: Is this shortcut valid for any number of terms?

The shortcut where the product of terms equals the middle term raised to the power of the number of terms is valid specifically for an odd number of terms in a G.P. In our case, we had $5$ terms, and the middle term was the $3$rd term. If you had $7$ terms, the product would be the $4$th term raised to the power of $7$. It works because the terms are symmetric around the middle term. If the number of terms is even, there is no single middle term, so the property changes slightly. Always check if the sequence has a clear center!

Question

The third term of a geometric progression is $4$. The product of the first five terms is

Options:

(A) $4^3$

(B) $4^5$

(C) $4^4$

(D) none of these

Sigma Percentile · Accepted Answer

### The Rhythm of Numbers: Unlocking the Geometric Progression

Welcome, future engineers! Today, we are going to dive into the elegant world of Geometric Progressions (G.P.).

Many students approach these problems as a chore of tedious calculation, but I want you to see them as a rhythmic dance of numbers. When you master the symmetry hidden within these sequences, you stop calculating and start seeing the architecture of the problem.

### Phase 1: Defining the Sequence

Imagine a sequence where each term is born from the previous one by multiplying it by a constant factor. This is the heartbeat of a Geometric Progression.

We define the first term as $a$ and the common ratio as $r$. Our sequence, therefore, unfolds as $a, ar, ar^2, ar^3, ar^4$.

This is our foundation. It is simple, it is clean, and it is powerful. Whenever you see a G.P., write these terms down immediately to clear the fog from your mind.

### Phase 2: The Golden Key

The problem provides us with a crucial piece of information: the third term, $T_3$, is equal to $4$. Looking at our sequence, the third term is clearly $ar^2$.

So, we have our golden key:
$$ar^2 = 4$$

Do not rush to solve for $a$ or $r$ individually. In JEE problems, often the path to the solution is not through finding the individual components, but through manipulating the expression as a whole. Keep this equation safe; it is the anchor for our entire calculation.

### Phase 3: The Algebraic Dance

Now, we need the product of the first five terms. Let us call this product $P$.

Writing it out, we get:
$$P = a \cdot (ar) \cdot (ar^2) \cdot (ar^3) \cdot (ar^4)$$

At first glance, this looks like a messy string of variables. But remember, multiplication is commutative. We can rearrange these terms however we like.

Let us group the $a$ terms and the $r$ terms separately. We have five $a$ terms, so that gives us $a^5$. For the $r$ terms, we have $r^1 \cdot r^2 \cdot r^3 \cdot r^4$.

Using the laws of indices, we add the exponents: $1 + 2 + 3 + 4 = 10$. Thus, our product simplifies to:
$$P = a^5 r^{10}$$

### Phase 4: The "Aha!" Moment

Here is where the magic happens. We have $P = a^5 r^{10}$, and we know $ar^2 = 4$.

Look at the exponents $5$ and $10$. They are both multiples of $5$. This is not a coincidence; it is the structural beauty of the G.P.

We can rewrite $a^5 r^{10}$ as $(a^1 r^2)^5$. Suddenly, the expression $ar^2$ appears right before our eyes! We substitute our golden key, $4$, into the expression:
$$P = (4)^5$$

### Conclusion: The Elegance of Symmetry

We have arrived at our answer: **$1024$** (or $4^5$). But more importantly, we have uncovered a general property.

For any G.P. with an odd number of terms, the product of those terms is always equal to the middle term raised to the power of the total number of terms. This is the kind of insight that separates the good from the elite.

JEE Advanced is not just about solving equations; it is about recognizing the patterns that make the equations solve themselves. Keep practicing this level of observation, and you will find that even the most intimidating problems start to look like simple, elegant puzzles.

Animated Solution for Mathematics - Sequence and Series: The third term of a geometric progression is $4$ . The product of the first five terms is

Select Answer:

Visualized Solution

Defining the G.P. Sequence

The Given Information

Setting up the Product

Grouping the Terms

Simplifying the $a$ Terms

Simplifying the $r$ Terms

Recognizing the Pattern

Substituting the Value

Final Conclusion

The Sigma Insight: Geometric Progression (G.P.)

The Rhythm of Numbers

Phase 1

Phase 2

Phase 3

Phase 4

Conclusion

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Animated Solution for Mathematics - Sequence and Series: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The third term of a geometric progression is 4. The product of the first five terms is