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Find the sum of 'n' terms of the series....

Find the sum of 'n' terms of the series.
`log_(2)(x/y) + log_(4)(x/y)^(2) + log_(8)(x/y)^(3) + log_(16)(x/y)^(4)` +……..

A

`log_(2)(x/y)^(4n)`

B

`n (log_(2)(x/y))`

C

`log_(2)(x^(n-1)/y^(n-1))`

D

`1/2 log_(2)(x/y)^(n(n+1))`

Text Solution

AI Generated Solution

The correct Answer is:
To find the sum of the series given by \[ S_n = \log_{2}(x/y) + \log_{4}((x/y)^{2}) + \log_{8}((x/y)^{3}) + \log_{16}((x/y)^{4}) + \ldots \] we will analyze each term in the series and then sum them up. ### Step 1: Rewrite each term using logarithmic properties We know that the logarithmic property states: \[ \log_{b}(a^n) = n \cdot \log_{b}(a) \] Using this property, we can rewrite the terms in the series: 1. The first term is \(\log_{2}(x/y)\). 2. The second term can be rewritten as: \[ \log_{4}((x/y)^{2}) = 2 \cdot \log_{4}(x/y) \] Since \(4 = 2^2\), we can convert the base: \[ \log_{4}(x/y) = \frac{1}{2} \cdot \log_{2}(x/y) \implies 2 \cdot \log_{4}(x/y) = 2 \cdot \frac{1}{2} \cdot \log_{2}(x/y) = \log_{2}(x/y) \] 3. The third term: \[ \log_{8}((x/y)^{3}) = 3 \cdot \log_{8}(x/y) \] Since \(8 = 2^3\), we convert the base: \[ \log_{8}(x/y) = \frac{1}{3} \cdot \log_{2}(x/y) \implies 3 \cdot \log_{8}(x/y) = 3 \cdot \frac{1}{3} \cdot \log_{2}(x/y) = \log_{2}(x/y) \] 4. The fourth term: \[ \log_{16}((x/y)^{4}) = 4 \cdot \log_{16}(x/y) \] Since \(16 = 2^4\), we convert the base: \[ \log_{16}(x/y) = \frac{1}{4} \cdot \log_{2}(x/y) \implies 4 \cdot \log_{16}(x/y) = 4 \cdot \frac{1}{4} \cdot \log_{2}(x/y) = \log_{2}(x/y) \] ### Step 2: Summing up the terms Now, we can see that each term simplifies to \(\log_{2}(x/y)\): \[ S_n = \log_{2}(x/y) + \log_{2}(x/y) + \log_{2}(x/y) + \ldots \text{ (n terms)} \] Thus, the sum of \(n\) terms is: \[ S_n = n \cdot \log_{2}(x/y) \] ### Final Answer The sum of the first \(n\) terms of the series is: \[ S_n = n \cdot \log_{2}(x/y) \]
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Knowledge Check

  • Find the sum of 'n' terms of the series. log_2(x/y)+log_4(x/y)^2+log_16(x/y)^4+... :

    A
    `log_2(x/y)^(4n)`
    B
    `n("log"_2(x)/y)`
    C
    `log_2(x^(n-1)/y^(n-1))`
    D
    `1/2log_2(x/y)^(n(n+1))`
  • If log_(4) x + log_(8)x^(2) + log_(16)x^(3) = (23)/(2) , then log_(x) 8 =

    A
    2
    B
    `(1)/(2)`
    C
    3
    D
    `(3)/(4)`
  • Find the value of x and y respectively for log_(10)(x^(2)y^(3))=7 and log_(10)(x/y)

    A
    x=10, y=100
    B
    x=100, y=10
    C
    x=10, y=20
    D
    none of these
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