QJEEM25Ma580f: JEE Main 2025 Mathematics - Class 11 JEEM25Ma580f Solution: Let be a G.P. of increasing positive terms. If and , then is…

First, let us denote the first term of the G.P. by $a_1 = A$ and the common ratio (which is > 1 , since the G.P. is increasing) by $r$ . Then the terms are:

$a_1 = A,\quad a_2 = Ar,\quad a_3 = Ar^2,\quad a_4 = Ar^3,\quad a_5 = Ar^4,\quad a_6 = Ar^5,\;\dots$

We are given:

$a_1 \cdot a_5 = 28$ , i.e.

$A \cdot (A r^4) \;=\; A^2 r^4 \;=\; 28. \quad (1)$

$a_2 + a_4 = 29$ , i.e.

$Ar \;+\; A r^3 \;=\; A(r + r^3) \;=\; 29. \quad (2)$

We want to find $a_6 = A r^5$ .

1. Solve for $r$

From $\text{(1)}$ :

$A^2 r^4 = 28 \quad\Longrightarrow\quad A^2 = \frac{28}{r^4}.$

From $\text{(2)}$ :

$A\,\bigl(r + r^3\bigr) = 29 \quad\Longrightarrow\quad A = \frac{29}{r + r^3}.$

Plug $A$ from $\text{(2)}$ into $\text{(1)}$ . After some algebra (or by a systematic approach), one finds that

$r^2 = 28 \quad\Longrightarrow\quad r = \sqrt{28} \;=\; 2\sqrt{7}$

(since r>1 , we take the positive root).

2. Solve for $A$

From (2), using $r = 2\sqrt{7}$ :

$r + r^3 = 2\sqrt{7} + (2\sqrt{7})^3 = 2\sqrt{7} + 56\sqrt{7} = 58\sqrt{7}.$

Hence,

$A = \frac{29}{\,58\sqrt{7}\,} = \frac{1}{2\sqrt{7}}.$

3. Find $a_6$

$a_6 = A\,r^5 = \frac{1}{2\sqrt{7}}\;\times\;(2\sqrt{7})^5.$

Compute $(2\sqrt{7})^5$ . First, $(2\sqrt{7})^2 = 28$ ; thus

$(2\sqrt{7})^5 = (2\sqrt{7})^4 \cdot (2\sqrt{7}) = 784 \times (2\sqrt{7}) = 1568\sqrt{7}.$

Therefore,

$a_6 = \frac{1}{2\sqrt{7}} \cdot 1568\sqrt{7} = \frac{1568}{2}\;\;(\text{since }\sqrt{7} \text{ cancels}) = 784.$

Answer: $a_6 = 784$ .

Hence, the correct option is Option B.