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