"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:"

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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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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:

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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:

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