JEE MAIN - Mathematics (2024 - 8th April Evening Shift - No. 11)

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 $$4^{\text {th }}, 6^{\text {th }}$$ and $$8^{\text {th }}$$ terms is equal to:
78
96
91
84

Explanation

Let's denote the first term of the geometric progression by $$a$$ and the common ratio by $$r$$. The terms of the geometric progression can be written as follows:

First term: $$a$$

Second term: $$ar$$

Third term: $$(ar^2)$$

Fourth term: $$(ar^3)$$

Fifth term: $$(ar^4)$$

Sixth term: $$(ar^5)$$

Eighth term: $$(ar^7)$$

We are given two key pieces of information:

1. The sum of the second and sixth terms is $$\frac{70}{3}$$:

$$ar + ar^5 = \frac{70}{3}$$

2. The product of the third and fifth terms is 49:

$$(ar^2) \cdot (ar^4) = 49$$

$$a^2 r^6 = 49$$

$$a^2 = \frac{49}{r^6}$$

$$a = \frac{7}{r^3}$$

Substituting $$a = \frac{7}{r^3}$$ into the first equation:

$$\frac{7}{r^3} \cdot r + \frac{7}{r^3} \cdot r^5 = \frac{70}{3}$$

$$\frac{7r}{r^3} + \frac{7r^5}{r^3} = \frac{70}{3}$$

$$\frac{7}{r^2} + \frac{7r^2}{1} = \frac{70}{3}$$

Let $$x = r^2$$. Then:

$$\frac{7}{x} + 7x = \frac{70}{3}$$

Multiply through by 3x to clear the denominator:

$$21 + 21x^2 = 70x$$

Rearrange into a standard quadratic equation:

$$21x^2 - 70x + 21 = 0$$

Divide by 7 to simplify:

$$3x^2 - 10x + 3 = 0$$

Solve this quadratic equation using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Where $$a = 3$$, $$b = -10$$, and $$c = 3$$. Thus:

$$x = \frac{10 \pm \sqrt{100 - 36}}{6}$$

$$x = \frac{10 \pm \sqrt{64}}{6}$$

$$x = \frac{10 \pm 8}{6}$$

$$x = 3$$ or $$x = \frac{1}{3}$$

Since $$x = r^2$$ and $$r$$ is positive, we get $$r = \sqrt{3}$$ or $$r = \frac{1}{\sqrt{3}}$$. We need to choose the value that results in positive, increasing terms:

If $$r = \sqrt{3}$$:

$$a = \frac{7}{r^3} = \frac{7}{(\sqrt{3})^3} = \frac{7}{3\sqrt{3}} = \frac{7}{3} \cdot \frac{1}{\sqrt{3}} = \frac{7\sqrt{3}}{9}$$

Now we can determine the sum of the 4th, 6th, and 8th terms:

The 4th term is: $$ar^3 = \frac{7\sqrt{3}}{9} \cdot 3\sqrt{3} = 7$$

The 6th term is: $$ar^5 = \frac{7\sqrt{3}}{9} \cdot 9\sqrt{3} = 21$$

The 8th term is: $$ar^7 = \frac{7\sqrt{3}}{9} \cdot 27(\sqrt{3}) = 49$$

Adding these together:

$$(4th + 6th + 8th terms) = 7 + 21 + 63 = 91$$

Therefore, the sum of the 4th, 6th, and 8th terms is 91.

Correct answer:

Option C: 91

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