In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 7

Q: Wait, why is $a_3 \cdot a_5$ equal to $(a_4)^2$? Is this always true?

Yes! In any Geometric Progression, the product of terms equidistant from a central term is equal to the square of that central term. Specifically, $a_n \cdot a_m = a_{n+k} \cdot a_{m-k}$. Here, $3+5 = 4+4$, so $a_3 \cdot a_5 = a_4 \cdot a_4 = (a_4)^2$. It is a powerful property that saves you from solving a system of equations with two variables.

Q: Why did we reject $t = 1/3$ when solving the quadratic? Is it possible for a GP to have a ratio less than 1?

A GP can definitely have a ratio less than 1 (a decreasing GP). However, the problem explicitly stated the GP is *increasing*. If $r^2 = 1/3$, then $r < 1$, which would make the sequence decrease. We must respect the constraints given in the problem statement. Always check your constraints!

Q: Do I need to find the first term $a$ and the common ratio $r$ separately to solve this?

That is a common trap! You *could* find them, but it's unnecessary labor. Notice how the expressions factor beautifully. In JEE, time is currency; look for these algebraic structures to save time and avoid messy calculations.

Question

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49. Then the sum of the $4^{	ext{th}}, 6^{	ext{th}}$ and $8^{	ext{th}}$ terms is equal to :

Options:

(A) 96

(B) 91

(C) 84

(D) 78

Sigma Percentile · Accepted Answer

### Analyzing the Setup

A Geometric Progression (GP) is defined by a first term $a$ and a common ratio $r$. The problem provides the product of the third and fifth terms: $a_3 \cdot a_5 = 49$.

Expressing these in terms of $a$ and $r$, we have:
$$(ar^2) \cdot (ar^4) = 49$$

This simplifies to:
$$a^2r^6 = 49$$

Notice the hidden symmetry: this expression is equivalent to $(ar^3)^2 = 49$. Since $ar^3$ represents the fourth term $a_4$, we have discovered that $a_4^2 = 49$. Given that the terms are positive, we conclude:
$$a_4 = 7$$

### The Algebraic Bridge

Next, we consider the sum of the second and sixth terms: $a_2 + a_6 = \frac{70}{3}$. Using the general term formula, this is:
$$ar + ar^5 = \frac{70}{3}$$

To connect this to our known value of $a_4$, we factor out $ar^3$:
$$ar^3 \left( \frac{1}{r^2} + r^2 \right) = \frac{70}{3}$$

Substituting $a_4 = 7$ into the equation, we get:
$$7 \left( \frac{1}{r^2} + r^2 \right) = \frac{70}{3}$$

Dividing both sides by 7, we arrive at the simplified equation:
$$\frac{1}{r^2} + r^2 = \frac{10}{3}$$

### The Quadratic Crossroads

To solve for $r$, let $t = r^2$. The equation transforms into:
$$t + \frac{1}{t} = \frac{10}{3}$$

Multiplying by $3t$ yields the quadratic equation:
$$3t^2 - 10t + 3 = 0$$

Factoring this quadratic gives $(3t - 1)(t - 3) = 0$, which results in $t = 3$ or $t = \frac{1}{3}$. Since the problem states the GP is **increasing**, we must have $r > 1$, which implies $r^2 > 1$. Therefore, we reject $t = \frac{1}{3}$ and accept:
$$r^2 = 3$$

### The Grand Finale

We are tasked with finding the sum of the fourth, sixth, and eighth terms: $S = a_4 + a_6 + a_8$. Writing this in terms of $a$ and $r$:
$$S = ar^3 + ar^5 + ar^7$$

Factoring out $ar^3$ reveals the structure:
$$S = ar^3(1 + r^2 + r^4)$$

We know $ar^3 = 7$, $r^2 = 3$, and consequently $r^4 = (r^2)^2 = 9$. Substituting these values:
$$S = 7(1 + 3 + 9) = 7(13) = 91$$

The final result is **91**.

Animated Solution for Mathematics - Sequence and Series: In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49. Then the sum of the $4^{th}, 6^{th}$ and $8^{th}$ terms is equal to :

Select Answer:

Visualized Solution

Define the Geometric Progression

Analyze the Product of $a_{3}$ and $a_{5}$

Find the Fourth Term $a_{4}$

Analyze the Sum of $a_{2}$ and $a_{6}$

Substitute and Simplify

Form and Solve the Quadratic Equation

Apply the Increasing Condition

Identify the Target Sum

Final Calculation

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

Analyzing the Setup

The Algebraic Bridge

The Quadratic Crossroads

The Grand Finale

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