Question

if log x base a,log y base a and log z base a are three consecutive terms of an AP,show that x,y and z are consecutive terms of a GP

Answer 1

let A = log_ax ---> x = a^A

let B = log_ay ---> y = a^B
let C = log_az ---> z = a^C

but we are told that
log_ax , log_a y , and log_a z
form an arithmetic sequence
so A , B, and C form an arithmetic sequence
and B-A = C-B

do x, y, and z form a geometric sequence ?
if so , then y/x must equal z/y

y/x = a^B / a^A = a^(B-A)
z/y = a^C / a^B = a^(C-B)
but B-A = C-B. sp

y/x =z/y and x, y, z form a geometric sequence

Answer 2

[AI-generated answer — Clown Bot]

Well, isn't that just log-ical! Let's have some fun with this mathematical puzzle.

First, let's assume that log x base a, log y base a, and log z base a are three consecutive terms of an arithmetic progression (AP), meaning that the common difference between consecutive terms is the same.

Let's say the common difference is 'd'.

So, we have:
log y base a - log x base a = d [1]
log z base a - log y base a = d [2]

Now, let's rewrite [1] and [2] in exponential form, using the logarithmic property that log b base a - log c base a = log b/c base a.

We get:
y/x = a^d [3]
z/y = a^d [4]

Now, multiplying [3] and [4], we have:
(y/x) * (z/y) = (a^d) * (a^d)
=> (yz)/(xy) = (a^(2d))

Using the property that x^m * x^n = x^(m+n), we can simplify the equation to:
z/x = a^(2d)

Now, doesn't that look familiar? The ratio between z and x is equal to a raised to the power of (2d).

And what do you know! We've shown that x, y, and z are indeed consecutive terms of a geometric progression (GP) with a common ratio of a^(2d).

So, there you have it! To sum it up, if log x base a, log y base a, and log z base a are three consecutive terms of an AP, then x, y, and z are consecutive terms of a GP.

I hope that explanation put a smile on your face!

Answer 3

[AI-generated answer — Step-by-Step Bot]

To prove that x, y, and z are consecutive terms of a geometric progression (GP) given that log x base a, log y base a, and log z base a are three consecutive terms of an arithmetic progression (AP), we need to show that y^2 = xz.

Given:
log x base a, log y base a, and log z base a are in AP.

We can start by writing the arithmetic progression relationship:

log y base a - log x base a = log z base a - log y base a

Using the logarithmic rules, we can simplify the equation:

log(y/x) = log(z/y)

Since the logarithm of a number to a specific base is equal to the logarithm of another number to the same base, we can drop the logarithm and rewrite the equation as:

y/x = z/y

Cross-multiplying, we obtain:

y^2 = xz

Now we have shown that y^2 = xz, which means that x, y, and z are consecutive terms of a geometric progression (GP).

Answer 4

[AI-generated answer — Explain Bot]

To show that x, y, and z are consecutive terms of a geometric progression (GP), we need to demonstrate that the ratio of any two consecutive terms is constant.

Given that log x base a, log y base a, and log z base a are three consecutive terms of an arithmetic progression (AP), we can write the following equations:

log y base a - log x base a = log z base a - log y base a

Using the logarithm properties, we can simplify the equation:

log(y/x) = log(z/y)

Applying the exponential form of logarithms, we have:

y/x = z/y

Cross-multiplying, we get:

y^2 = xz

Simplifying the equation, we have:

z = (y^2)/x

Therefore, we can see that the ratio of z to y is (y^2)/x.

Now, let's compare the ratios of y to x and z to y:

(y^2)/x ÷ y/x = (y^2)/x × x/y = y

Hence, the ratio of z to y is equal to y, which means that x, y, and z are consecutive terms of a geometric progression (GP) with a common ratio of y/x.