Question

If log x, log, x and log x be in HP, then ab and c are in
(a) AP
(b) HP
(c) GP
(d) None of these
o in​

Answer 1

Answer:

If $log_{a}x$, $log_{b}x$, and $log_{c}x$ are in Harmonic Progression, then a, b, and c are in (c) GP.

Geometric Progression:

• Geometric progression is a sequence in which the ratio of any two corresponding terms is always a constant.
• It can also be said that excluding the first term, all other terms of a geometric progression can be calculated by multiplying the previous term by a fixed number.
• The sequence is of the form $a, ar^{2}, ar^{3}, ar^{4}, ...$
• The terms of a geometric progression are obtained by the formula $a_{n}=a_{1}r^{n-1}$, where $a_{1}$ is the first term of the sequence and r is the common ratio.

Example: The sequence $4, 16, 64, 256, ...$ is a geometric progression with a common ratio of 4.

Step-by-step explanation:

The given sequence $log_{a}x$, $log_{b}x$, $log_{c}x$ is a harmonic progression.

Step 1:

As a harmonic progression is the reciprocal of an arithmetic progression, the arithmetic mean of the first and last terms of the given harmonic progression will be equal to the second term.

$\frac{2}{ log_{b}x}=\frac{1}{ log_{a}x} + \frac{1}{ log_{c}x}$

$2log_{x}b= log_{x}a + log_{x}c$

Step 2:

Separating the base of the logarithms, we get

$2\frac{logb}{logx}= \frac{loga}{logx} + \frac{log c}{logx}$

$logx$ being common on both sides gets canceled out.

$2logb=log a + log c$

Step 3:

Applying logarithm formulas and simplifying.

$logb^{2} = loga+logc$

$logb^{2}=log(ac)$

$b^{2} = ac$

$\frac{b}{a}=\frac{c}{b}$

The three terms are said to be in geometric progression if the ratio of the consecutive terms is constant.

Explanation for incorrect options:

(a) AP: Arithmetic progression is a sequence or series in which the difference between any two consecutive terms in the series is always constant. The terms of an arithmetic progression are obtained by the formula $a_{n}=a_{1}+(n-1)d$, where $a_{1}$ is the first term of the sequence and d is the common difference.

(b) HP: Harmonic progression is a sequence or series which is obtained by taking the reciprocals of the arithmetic progression. The terms of a harmonic progression are obtained by the formula $a_{n}=\frac{1}{a+(n-1)d}$, where a is the first term of the sequence and d is the common difference.

Therefore, a, b, and c are in geometric progression (GP).

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