JEE Main 2024 — Sequences And Series Question with Solution

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: $$(a{r}^2)$$

Fourth term: $$(a{r}^3)$$

Fifth term: $$(a{r}^4)$$

Sixth term: $$(a{r}^5)$$

Eighth term: $$(a{r}^7)$$

We are given two key pieces of information:

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

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

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

$$(a{r}^2)\cdot (a{r}^4)=49$$

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

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

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

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: $$a{r}^3=\frac{7\sqrt{3}}{9}\cdot 3\sqrt{3}=7$$

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

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

Adding these together:

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

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

Correct answer:

Option C: 91