JEE Main 2021 (26 February Shift 1)

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Animated Solution for Mathematics - Sequence and Series: In an increasing, geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is 25. Then, the sum of $4^{t h}, 6^{t h}$ and $8^{t h}$ terms is equal to :

Q: Is there a faster way to find the sum of the $4^{th}, 6^{th},$ and $8^{th}$ terms?

Yes! Instead of finding $a$ and $r$ individually, notice that the sum $T_4 + T_6 + T_8$ can be factored as $ar^3(1 + r^2 + r^4)$. Since we already know $ar^3 = 5$ and $r^2 = 2$, we can compute the sum directly without ever needing to find the exact value of $a$ or $r$ separately. This is a classic JEE efficiency trick.

Select Answer:

Visualized Solution

Defining the Geometric Series

Let the first term of the G.P. be $a$ .
Let the common ratio be $r$ .
The terms of the G.P. are: $a, a r, a r^{2}, a r^{3}, a r^{4}, a r^{5}, \dots$

Translating the First Condition

Condition 1: Sum of $2^{n d}$ and $6^{t h}$ term is $\frac{25}{2}$ .
$T_{2} + T_{6} = \frac{25}{2}$
$a r + a r^{5} = \frac{25}{2} \dots (1)$

Analyzing the Product Condition

Condition 2: Product of $3^{r d}$ and $5^{t h}$ term is $25$ .
$T_{3} \cdot T_{5} = 25$
$(a r^{2}) \cdot (a r^{4}) = 25$
$a^{2} r^{6} = 25$

Finding the Mid-term Value

Taking the square root: $a r^{3} = 5 \dots (2)$
Since the series is increasing, terms are positive, so we take the positive root.
From this, we can write $a = \frac{5}{r ^{3}}$ .

Substitution and Simplification

Substitute $a = \frac{5}{r ^{3}}$ into equation (1):
$(\frac{5}{r ^{3}}) r + (\frac{5}{r ^{3}}) r^{5} = \frac{25}{2}$
$\frac{5}{r ^{2}} + 5 r^{2} = \frac{25}{2}$

Forming the Quadratic Equation

Divide by $5$ : $\frac{1}{r ^{2}} + r^{2} = \frac{5}{2}$
Let $r^{2} = t$ . Then: $\frac{1}{t} + t = \frac{5}{2}$
Multiply by $2 t$ : $2 + 2 t^{2} = 5 t$
$2 t^{2} - 5 t + 2 = 0$

Solving for the Common Ratio

Factorizing the quadratic: $(2 t - 1) (t - 2) = 0$
Possible values for $t$ : $t = \frac{1}{2}$ or $t = 2$ .
Since the series is increasing, $r^{2} > 1$ .
Therefore, $t = r^{2} = 2$ .

Setting up the Final Goal

Target: Sum of $4^{t h}, 6^{t h}$ and $8^{t h}$ terms.
Sum $= a r^{3} + a r^{5} + a r^{7}$
Factor out $a r^{3}$ : Sum $= a r^{3} (1 + r^{2} + r^{4})$

The Final Calculation

Substitute $a r^{3} = 5$ and $r^{2} = 2$ :
Sum $= 5 (1 + 2 + 2^{2})$
Sum $= 5 (1 + 2 + 4)$
Sum $= 5 (7) = 35$

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

Analyzing the Geometric DNA

A geometric series is defined by its first term

a

and common ratio

r

. The sequence of terms is given by

a, a r, a r^{2}, a r^{3}, a r^{4}, a r^{5}, \dots

Our objective is to determine the sum of the

4^{t h}

6^{t h}

, and

8^{t h}

terms, which corresponds to the expression

a r^{3} + a r^{5} + a r^{7}

Decoding the Clues

We are provided with two primary constraints. First, the sum of the

2^{n d}

and

6^{t h}

terms is

\frac{25}{2}

a r + a r^{5} = \frac{25}{2}

Second, the product of the

3^{r d}

and

5^{t h}

terms is

25

(a r^{2}) \cdot (a r^{4}) = 25 \Rightarrow a^{2} r^{6} = 25

Recognizing that

a^{2} r^{6} = (a r^{3})^{2}

, we take the square root to find the "golden key":

a r^{3} = 5

The Quadratic Dance

Given

a r^{3} = 5

, we express the first term as

a = \frac{5}{r ^{3}}

. Substituting this into our first equation yields:

(\frac{5}{r ^{3}}) r + (\frac{5}{r ^{3}}) r^{5} = \frac{25}{2}

Simplifying the expression, we obtain:

\frac{5}{r ^{2}} + 5 r^{2} = \frac{25}{2} \Rightarrow \frac{1}{r ^{2}} + r^{2} = \frac{5}{2}

Letting

t = r^{2}

, we solve the resulting quadratic equation:

\frac{1}{t} + t = \frac{5}{2} \Rightarrow 2 t^{2} - 5 t + 2 = 0

Factoring the quadratic gives

(2 t - 1) (t - 2) = 0

, yielding

t = \frac{1}{2}

t = 2

. Since the series is increasing, we must have

r > 1

, which implies

r^{2} > 1

. Thus, we select

t = 2

The Final Crescendo

We now calculate the sum of the

4^{t h}

6^{t h}

, and

8^{t h}

terms:

T_{4} + T_{6} + T_{8} = a r^{3} + a r^{5} + a r^{7}

Factoring out

a r^{3}

, we get:

a r^{3} (1 + r^{2} + r^{4})

Substituting the known values

a r^{3} = 5

and

r^{2} = 2

5 (1 + 2 + 2^{2}) = 5 (1 + 2 + 4) = 5 (7) = 35

The final result is 35.

Common Traps & FAQs

Wait, why do we reject the root

t = \frac{1}{2}

Great catch! The problem explicitly states that the geometric series is increasing. For a geometric series with positive terms to be increasing, the common ratio

r

must be greater than

1

. Since

t = r^{2}

, this implies

t

must be greater than

1

. Thus, we reject

t = \frac{1}{2}

and accept

t = 2

Why can't we just use energy conservation or some other physics principle?

This is a pure mathematics problem involving sequences and series, not a physics problem. In JEE, you will encounter both. Here, we rely on the algebraic properties of geometric progressions. Think of the G.P. terms as the 'coordinates' of our sequence; we are simply navigating them using the rules of exponents.

Is there a faster way to find the sum of the

4^{t h}, 6^{t h},

and

8^{t h}

terms?

Yes! Instead of finding

a

and

r

individually, notice that the sum

T_{4} + T_{6} + T_{8}

can be factored as

a r^{3} (1 + r^{2} + r^{4})

. Since we already know

a r^{3} = 5

and

r^{2} = 2

, we can compute the sum directly without ever needing to find the exact value of

a

r

separately. This is a classic JEE efficiency trick.

Animated Solution for Mathematics - Sequence and Series: In an increasing, geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is 25. Then, the sum of $4^{t h}, 6^{t h}$ and $8^{t h}$ terms is equal to :

Select Answer:

Visualized Solution

Defining the Geometric Series

Translating the First Condition

Analyzing the Product Condition

Finding the Mid-term Value

Substitution and Simplification

Forming the Quadratic Equation

Solving for the Common Ratio

Setting up the Final Goal

The Final Calculation

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

Analyzing the Geometric DNA

Decoding the Clues

The Quadratic Dance

The Final Crescendo

Similar Questions

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Select Answer:

Visualized Solution

Defining the Geometric Series

Translating the First Condition

Analyzing the Product Condition

Finding the Mid-term Value

Substitution and Simplification

Forming the Quadratic Equation

Solving for the Common Ratio

Setting up the Final Goal

The Final Calculation

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

Analyzing the Geometric DNA

Decoding the Clues

The Quadratic Dance

The Final Crescendo

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