JEE Main 2020 (5 Sep Evening)

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Animated Solution for Mathematics - Sequence and Series: 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

Q: Can I just add the numbers $2+3+4+\dots+21$?

Yes, absolutely! Once you factor out $\log_7 x$, you are left with $\log_7 x \times (2 + 3 + 4 + \dots + 21)$. The sum of the first $n$ integers is $\frac{n(n+1)}{2}$. You can use this to find the sum of the sequence $2, 3, \dots, 21$ and then multiply by $\log_7 x$. It is a very elegant way to verify your result.

Select Answer:

Visualized Solution

The Given Logarithmic Series

Given series: $lo g_{(7^{1/2})} x + lo g_{(7^{1/3})} x + lo g_{(7^{1/4})} x + \dots$
Number of terms: $n = 20$
Sum of the first 20 terms: $S_{20} = 460$

Logarithm Base Power Property

Recall the property: $lo g_{b^{k}} a = \frac{1}{k} lo g_{b} a$
The exponent in the base comes out as its reciprocal.

Simplifying the First Term

First term: $lo g_{(7^{1/2})} x$
Apply the property: $k = \frac{1}{2}$
$\frac{1}{1/2} lo g_{7} x = 2 lo g_{7} x$

Simplifying the Next Terms

Second term: $lo g_{(7^{1/3})} x = \frac{1}{1/3} lo g_{7} x = 3 lo g_{7} x$
Third term: $lo g_{(7^{1/4})} x = \frac{1}{1/4} lo g_{7} x = 4 lo g_{7} x$

The New Series Structure

Simplified series: $2 lo g_{7} x + 3 lo g_{7} x + 4 lo g_{7} x + \dots$
This continues for $20$ terms.

Identifying the A.P.

The terms form an Arithmetic Progression (A.P.)
First term: $a = 2 lo g_{7} x$
Common difference: $d = 3 lo g_{7} x - 2 lo g_{7} x = lo g_{7} x$
Number of terms: $n = 20$

Sum of an A.P.

Sum formula: $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
We need to find $S_{20}$

Substituting into the Formula

Substitute $n = 20$ , $a = 2 lo g_{7} x$ , $d = lo g_{7} x$
$S_{20} = \frac{20}{2} [2 (2 lo g_{7} x) + (20 - 1) (lo g_{7} x)]$

Simplifying the Expression

$S_{20} = 10 [4 lo g_{7} x + 19 lo g_{7} x]$
Combine the like terms inside the bracket.

The Final Sum Expression

$S_{20} = 10 [23 lo g_{7} x]$
$S_{20} = 230 lo g_{7} x$

Equating to the Given Value

We are given that $S_{20} = 460$
Therefore, $230 lo g_{7} x = 460$

Isolating the Logarithm

Divide both sides by $230$ :
$lo g_{7} x = \frac{460}{230}$
$lo g_{7} x = 2$

Converting to Exponential Form

By definition: if $lo g_{b} a = c$ , then $a = b^{c}$
$x = 7^{2}$
$x = 49$

00:00 / 00:00

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

Analyzing the Logarithmic Series

We are given the series:

lo g_{(7^{1/2})} x + lo g_{(7^{1/3})} x + lo g_{(7^{1/4})} x + \dots

The base of each logarithm is a fractional power of

7

. To simplify this, we invoke the fundamental property of logarithms:

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

Applying the Property

When we apply this property to the first term,

lo g_{(7^{1/2})} x

, the exponent

k = \frac{1}{2}

moves to the front as its reciprocal, which is

2

. Thus, the first term becomes

2 lo g_{7} x

Following the same logic for the second term,

lo g_{(7^{1/3})} x

, the exponent

\frac{1}{3}

yields a reciprocal of

3

. Consequently, the second term becomes

3 lo g_{7} x

The pattern is now clear. The series can be rewritten as:

2 lo g_{7} x + 3 lo g_{7} x + 4 lo g_{7} x + \dots

Summing the Arithmetic Progression

This is an Arithmetic Progression (A.P.) consisting of

20

terms. Here, the first term

a = 2 lo g_{7} x

and the common difference

d = lo g_{7} x

The sum of an A.P. is given by the formula:

S_{n} = \frac{n}{2} [2 a + (n - 1) d]

Substituting our values for

n = 20

S_{20} = \frac{20}{2} [2 (2 lo g_{7} x) + (20 - 1) (lo g_{7} x)]

Final Calculation

Simplifying the expression inside the brackets:

S_{20} = 10 [4 lo g_{7} x + 19 lo g_{7} x]

S_{20} = 10 [23 lo g_{7} x] = 230 lo g_{7} x

We are given that the sum equals

460

. Therefore:

230 lo g_{7} x = 460

Dividing both sides by

230

, we obtain:

lo g_{7} x = 2

By the definition of logarithms, we find the final value: $x = 7^{2} = 49$

Common Traps & FAQs

Why does the exponent in the base flip when it comes out?

Think of the logarithm as asking 'to what power must I raise the base to get the argument?' If the base is already a power, say

b^{k}

, then to get

a

, you need to raise

b^{k}

to some power

y

. So

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

, which means

b^{k y} = a

. Thus

k y = lo g_{b} a

, or

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

. The reciprocal is a natural consequence of the exponent laws.

How do I know it's an AP?

An Arithmetic Progression is defined by a constant difference between consecutive terms. After simplifying, the terms are

2 lo g_{7} x, 3 lo g_{7} x, 4 lo g_{7} x, \dots

. The difference between any two consecutive terms is

(k + 1) lo g_{7} x - k lo g_{7} x = lo g_{7} x

, which is constant. That is the definition of an AP.

Can I just add the numbers

2 + 3 + 4 + \dots + 21

Yes, absolutely! Once you factor out

lo g_{7} x

, you are left with

lo g_{7} x \times (2 + 3 + 4 + \dots + 21)

. The sum of the first

n

integers is

\frac{n ( n + 1 )}{2}

. You can use this to find the sum of the sequence

2, 3, \dots, 21

and then multiply by

lo g_{7} x

. It is a very elegant way to verify your result.

Animated Solution for Mathematics - Sequence and Series: 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

Select Answer:

Visualized Solution

The Given Logarithmic Series

Logarithm Base Power Property

Simplifying the First Term

Simplifying the Next Terms

The New Series Structure

Identifying the A.P.

Sum of an A.P.

Substituting into the Formula

Simplifying the Expression

The Final Sum Expression

Equating to the Given Value

Isolating the Logarithm

Converting to Exponential Form

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

Analyzing the Logarithmic Series

Applying the Property

Summing the Arithmetic Progression

Final Calculation

Similar Questions

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

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Select Answer:

Visualized Solution

The Given Logarithmic Series

Logarithm Base Power Property

Simplifying the First Term

Simplifying the Next Terms

The New Series Structure

Identifying the A.P.

Sum of an A.P.

Substituting into the Formula

Simplifying the Expression

The Final Sum Expression

Equating to the Given Value

Isolating the Logarithm

Converting to Exponential Form

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

Analyzing the Logarithmic Series

Applying the Property

Summing the Arithmetic Progression

Final Calculation

Similar Questions

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