JEE Main 2021 (20 July Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Sequence and Series: If sum of the first 21 terms of the series $lo g_{9^{1/2}} x + lo g_{9^{1/3}} x + lo g_{9^{1/4}} x + \dots$ , where $x > 0$ is 504, then $x$ is equal to

Q: Why does the exponent in the base come out as a reciprocal? Shouldn't it just come out as $k$?

Great question! Think about the definition. If $y = \log_{a^k} x$, then $(a^k)^y = x$, so $a^{ky} = x$. Taking $\log_a$ on both sides, $ky = \log_a x$, so $y = \frac{1}{k} \log_a x$. The reciprocal is a direct consequence of the base being raised to a power!

Q: How do I know the last term is 22? I thought there were 21 terms, so the last term should be 21.

It's a common trap! Look at the pattern: Term 1 has base $9^{1/2}$ (coefficient 2), Term 2 has base $9^{1/3}$ (coefficient 3). The coefficient is always $n+1$ where $n$ is the term index. For the 21st term, the coefficient is $21+1 = 22$.

Select Answer:

Visualized Solution

Analyzing the Logarithmic Series

Given series: $lo g_{9^{\frac{1}{2}}} x + lo g_{9^{\frac{1}{3}}} x + lo g_{9^{\frac{1}{4}}} x + \dots$
Total terms: $n = 21$
Sum of series: $S_{21} = 504$

The Log Base Power Rule

Logarithm Property:
$lo g_{a^{k}} m = \frac{1}{k} lo g_{a} m$
This property allows us to bring the exponent of the base to the front as its reciprocal.

Transforming the First Term

First term: $lo g_{9^{\frac{1}{2}}} x$
Using $\frac{1}{\frac{1}{2}} = 2$ :
Term 1 = $2 lo g_{9} x$

Transforming Subsequent Terms

Second term: $lo g_{9^{\frac{1}{3}}} x \to 3 lo g_{9} x$
Third term: $lo g_{9^{\frac{1}{4}}} x \to 4 lo g_{9} x$

Writing the Simplified Series

Simplified Series:
$S = (2 + 3 + 4 + \dots up to 21 terms) lo g_{9} x$

Identifying the Arithmetic Progression

The coefficients form an A.P.:
First term ( $a$ ) = $2$
Common difference ( $d$ ) = $1$
Number of terms ( $n$ ) = $21$

Finding the Last Term

$l = a + (n - 1) d$
$l = 2 + (21 - 1) 1$
$l = 2 + 20 = 22$

Applying the AP Sum Formula

Sum of A.P. formula: $S_{n} = \frac{n}{2} (a + l)$
$S_{21} = \frac{21}{2} (2 + 22)$

Calculating the Sum of Coefficients

$S_{21} = \frac{21}{2} \times 24$
$S_{21} = 21 \times 12 = 252$
Total Sum = $252 lo g_{9} x$

Setting up the Final Equation

Given: Sum = $504$
Equation: $252 lo g_{9} x = 504$

Solving for the Logarithm

$lo g_{9} x = \frac{504}{252}$
$lo g_{9} x = 2$

Converting to Exponential Form

Definition of Logarithm:
If $lo g_{b} a = c$ , then $a = b^{c}$
Here, $x = 9^{2}$

The Final Result

$x = 9 \times 9$
$x = 81$

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The Sigma Insight: Arithmetic Progression (A.P.)

Solution Diagram

Analyzing the Setup

We are tasked with solving for

x

in the following logarithmic series:

lo g_{9^{1/2}} x + lo g_{9^{1/3}} x + lo g_{9^{1/4}} x + \dots = 504

The sum consists of the first 21 terms. To simplify this, we utilize the Log Base Power Rule, which states that

lo g_{a^{k}} m = \frac{1}{k} lo g_{a} m

Simplifying the Terms

Applying this rule to each term, we observe the following transformations: - The first term:

lo g_{9^{1/2}} x = 2 lo g_{9} x

- The second term:

lo g_{9^{1/3}} x = 3 lo g_{9} x

- The third term:

lo g_{9^{1/4}} x = 4 lo g_{9} x

By factoring out the common term

lo g_{9} x

, the expression becomes:

(2 + 3 + 4 + \dots + 21 terms) lo g_{9} x = 504

Evaluating the Arithmetic Progression

The coefficients

2, 3, 4, \dots

form an Arithmetic Progression (AP) where the first term

a = 2

, the common difference

d = 1

, and the number of terms

n = 21

First, we determine the 21st term (

l

) using the formula

l = a + (n - 1) d

l = 2 + (21 - 1) 1 = 22

Next, we calculate the sum of these 21 terms using the formula

S_{n} = \frac{n}{2} (a + l)

S_{21} = \frac{21}{2} (2 + 22) = \frac{21}{2} (24) = 21 \times 12 = 252

Final Calculation

Substituting the sum back into our simplified equation, we obtain:

252 lo g_{9} x = 504

Dividing both sides by 252 yields:

lo g_{9} x = 2

Applying the fundamental definition of a logarithm, where

lo g_{b} a = c

implies

a = b^{c}

, we find:

x = 9^{2}

$x = 81$

Common Traps & FAQs

Why does the exponent in the base come out as a reciprocal? Shouldn't it just come out as

k

Great question! Think about the definition. If

y = lo g_{a^{k}} x

, then

(a^{k})^{y} = x

, so

a^{k y} = x

. Taking

lo g_{a}

on both sides,

k y = lo g_{a} x

, so

y = \frac{1}{k} lo g_{a} x

. The reciprocal is a direct consequence of the base being raised to a power!

How do I know the last term is 22? I thought there were 21 terms, so the last term should be 21.

It's a common trap! Look at the pattern: Term 1 has base

9^{1/2}

(coefficient 2), Term 2 has base

9^{1/3}

(coefficient 3). The coefficient is always

n + 1

where

n

is the term index. For the 21st term, the coefficient is

21 + 1 = 22

Can we use properties of logs to combine the terms first?

While you could try to combine them, it would lead to a very complex product inside the logarithm. It is much more elegant to factor out the common

lo g_{9} x

and treat the coefficients as a separate arithmetic series.

Animated Solution for Mathematics - Sequence and Series: If sum of the first 21 terms of the series $lo g_{9^{1/2}} x + lo g_{9^{1/3}} x + lo g_{9^{1/4}} x + \dots$ , where $x > 0$ is 504, then $x$ is equal to

Select Answer:

Visualized Solution

Analyzing the Logarithmic Series

The Log Base Power Rule

Transforming the First Term

Transforming Subsequent Terms

Writing the Simplified Series

Identifying the Arithmetic Progression

Finding the Last Term

Applying the AP Sum Formula

Calculating the Sum of Coefficients

Setting up the Final Equation

Solving for the Logarithm

Converting to Exponential Form

The Final Result

The Sigma Insight: Arithmetic Progression (A.P.)

Analyzing the Setup

Simplifying the Terms

Evaluating the Arithmetic Progression

Final Calculation

Similar Questions

If the sum of the first 20 terms of the series $lo g_{(7^{1/2})} x + lo g_{(7^{1/3})} x + lo g_{(7^{1/4})} x + ...$ is 460 then $x$ is equal to

If $lo g_{3} 2, lo g_{3} (2^{x} - 5), lo g_{3} (2^{x} - \frac{7}{2})$ are in an arithmetic progression, then the value of $x$ is equal to

If the sum of the first 40 terms of the series, $3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + \dots$ is $(102) m$ , then m is equal to :

If $lo g_{3} 2, lo g_{3} (2^{x} - 5)$ , and $lo g_{3} (2^{x} - \frac{7}{2})$ are in arithmetic progression, determine the value of $x$ .

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Let $a_{1}, a_{2}, ..., a_{2024}$ be an Arithmetic Progression such that $a_{1} + (a_{5} + a_{10} + a_{15} + ... + a_{2020}) + a_{2024} = 2233$ . Then $a_{1} + a_{2} + a_{3} + ... + a_{2024}$ is equal to

Suppose $a_{1}, a_{2}, \dots, a_{n}, \dots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is $5 : 17$ and $110 < a_{15} < 120$ , then the sum of the first ten terms of the progression is equal to -

Let 3, 6, 9, 12, ... upto 78 terms and 5, 9, 13, 17, ... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ______.

If the sum of the series $20 + 19 \frac{3}{5} + 19 \frac{1}{5} + 18 \frac{4}{5} + ...$ upto $n^{t h}$ term is 488 and the $n^{t h}$ term is negative, then

Let $a, b, c > 1, a^{'}, b^{'}$ and $c^{'}$ be in A.P., and $lo g_{a} b, lo g_{b} a$ and $lo g_{c} b$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a + 4 b + c}{3}$ and the common difference is $\frac{a - 8 b + c}{10}$ is $- 444$ , then $ab c$ is equal to

Select Answer:

Visualized Solution

Analyzing the Logarithmic Series

The Log Base Power Rule

Transforming the First Term

Transforming Subsequent Terms

Writing the Simplified Series

Identifying the Arithmetic Progression

Finding the Last Term

Applying the AP Sum Formula

Calculating the Sum of Coefficients

Setting up the Final Equation

Solving for the Logarithm

Converting to Exponential Form

The Final Result

The Sigma Insight: Arithmetic Progression (A.P.)

Analyzing the Setup

Simplifying the Terms

Evaluating the Arithmetic Progression

Final Calculation

Similar Questions

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If $lo g_{3} 2, lo g_{3} (2^{x} - 5)$ , and $lo g_{3} (2^{x} - \frac{7}{2})$ are in arithmetic progression, determine the value of $x$ .

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