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Animated Solution for Mathematics - Sequence and Series: The first two terms of a geometric progression add up to $12$ . the sum of the third and the fourth terms is $48$ . If the terms of the geometric progression are alternately positive and negative, then the first term is

Q: Wait, why do we reject the positive root for $r$?

Great question! The problem states the terms are alternately positive and negative. If $r$ were positive (i.e., $r=2$), then multiplying the first term $a$ by $r$ would keep the sign the same (e.g., $-12, -24, -48, \dots$). Only a negative common ratio ($r=-2$) forces the sequence to flip signs with every term, satisfying the condition.

Q: Can we solve this without dividing the equations?

You could try substitution, but dividing is the 'elegant' path. By dividing, we immediately cancel out the unknowns $a$ and $(1+r)$, reducing a complex system to a simple quadratic equation. It saves time and minimizes the chance of algebraic errors—a key skill for JEE Advanced!

Q: What if the sum of the first two terms was zero?

If $a+ar=0$, then $a(1+r)=0$. This would mean either $a=0$ (a trivial sequence) or $r=-1$. If $r=-1$, the sequence would be $a, -a, a, -a, \dots$, which also alternates in sign. Always check if your derived $r$ makes the denominator in your division step zero!

Select Answer:

Visualized Solution

Define the G.P. Terms

Let the first term of the G.P. be $a$ and the common ratio be $r$ .
The first four terms are: $T_{1} = a$ , $T_{2} = a r$ , $T_{3} = a r^{2}$ , $T_{4} = a r^{3}$ .

First Condition: Sum of First Two Terms

Given: $T_{1} + T_{2} = 12$
$a + a r = 12$
Factoring out $a$ : $a (1 + r) = 12$ --- (Equation 1)

Second Condition: Sum of Next Two Terms

Given: $T_{3} + T_{4} = 48$
$a r^{2} + a r^{3} = 48$
Factoring out $a r^{2}$ : $a r^{2} (1 + r) = 48$ --- (Equation 2)

Dividing the Equations

To eliminate $a$ , divide Equation 2 by Equation 1:
$\frac{a r ^{2} ( 1 + r )}{a ( 1 + r )} = \frac{48}{12}$

Solving for $r^{2}$

Cancel out $a (1 + r)$ from numerator and denominator.
$r^{2} = 4$

Analyzing the Sign of $r$

Since $r^{2} = 4$ , we have two possibilities: $r = 2$ or $r = - 2$ .
The problem states the terms are alternately positive and negative.

Concluding the Common Ratio

If $r > 0$ , all terms have the same sign.
If $r < 0$ , terms alternate in sign.
Therefore, $r = - 2$ .

Substituting $r$ to Find $a$

Substitute $r = - 2$ into Equation 1: $a (1 + r) = 12$
$a (1 + (- 2)) = 12$

Calculating the First Term

$a (1 - 2) = 12$
$a (- 1) = 12$
$a = - 12$

Visualizing the Alternating Sequence

The first term is $a = - 12$ .
The sequence is: $- 12, 24, - 48, 96, \dots$
The terms correctly alternate in sign!

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The Sigma Insight: Geometric Progression (G.P.)

Solution Diagram

Analyzing the Setup

In a Geometric Progression (G.P.), we define the terms using the first term

a

and the common ratio

r

. The sequence is represented as

a, a r, a r^{2}, a r^{3}, \dots

The problem provides two specific conditions based on the sum of consecutive terms. The first condition is

T_{1} + T_{2} = 12

, which translates to:

a + a r = 12 ⟹ a (1 + r) = 12 (Equation 1)

The second condition is

T_{3} + T_{4} = 48

, which translates to:

a r^{2} + a r^{3} = 48 ⟹ a r^{2} (1 + r) = 48 (Equation 2)

The Algebraic Symphony

To solve this system, we divide Equation 2 by Equation 1 to eliminate the variables

a

and

(1 + r)

\frac{a r ^{2} ( 1 + r )}{a ( 1 + r )} = \frac{48}{12}

This simplification yields:

r^{2} = 4

The Sign Trap

The equation

r^{2} = 4

yields two potential solutions:

r = 2

r = - 2

. We must apply the constraint provided in the problem statement: the terms alternate in sign.

r = 2

, the terms would maintain the same sign as

a

. However, if

r = - 2

, the sequence becomes

a, - 2 a, 4 a, - 8 a, \dots

, which perfectly satisfies the alternating sign condition. Therefore, we conclude that $r = - 2$ .

Final Calculation

With the value of

r

determined, we substitute it back into Equation 1 to find the first term

a

a (1 + (- 2)) = 12

a (- 1) = 12

a = - 12

We verify the result: the sequence is

- 12, 24, - 48, 96, \dots

. The sum of the first two terms is

- 12 + 24 = 12

, and the sum of the next two terms is

- 48 + 96 = 48

The first term of the progression is $- 12$ .

Common Traps & FAQs

Wait, why do we reject the positive root for

r

Great question! The problem states the terms are alternately positive and negative. If

r

were positive (i.e.,

r = 2

), then multiplying the first term

a

r

would keep the sign the same (e.g.,

- 12, - 24, - 48, \dots

). Only a negative common ratio (

r = - 2

) forces the sequence to flip signs with every term, satisfying the condition.

Can we solve this without dividing the equations?

You could try substitution, but dividing is the 'elegant' path. By dividing, we immediately cancel out the unknowns

a

and

(1 + r)

, reducing a complex system to a simple quadratic equation. It saves time and minimizes the chance of algebraic errors—a key skill for JEE Advanced!

What if the sum of the first two terms was zero?

a + a r = 0

, then

a (1 + r) = 0

. This would mean either

a = 0

(a trivial sequence) or

r = - 1

. If

r = - 1

, the sequence would be

a, - a, a, - a, \dots

, which also alternates in sign. Always check if your derived

r

makes the denominator in your division step zero!

Animated Solution for Mathematics - Sequence and Series: The first two terms of a geometric progression add up to $12$ . the sum of the third and the fourth terms is $48$ . If the terms of the geometric progression are alternately positive and negative, then the first term is

Select Answer:

Visualized Solution

Define the G.P. Terms

First Condition: Sum of First Two Terms

Second Condition: Sum of Next Two Terms

Dividing the Equations

Solving for $r^{2}$

Analyzing the Sign of $r$

Concluding the Common Ratio

Substituting $r$ to Find $a$

Calculating the First Term

Visualizing the Alternating Sequence

The Sigma Insight: Geometric Progression (G.P.)

Analyzing the Setup

The Algebraic Symphony

The Sign Trap

Final Calculation

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Select Answer:

Visualized Solution

Define the G.P. Terms

First Condition: Sum of First Two Terms

Second Condition: Sum of Next Two Terms

Dividing the Equations

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } r2

Analyzing the Sign of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } r

Concluding the Common Ratio

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } r to Find .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Calculating the First Term

Visualizing the Alternating Sequence

The Sigma Insight: Geometric Progression (G.P.)

Analyzing the Setup

The Algebraic Symphony

The Sign Trap

Final Calculation

Similar Questions

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