Verify if the sequence where loga, log(a^2/b), log(a^3/b^2), ... is an A.P.
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LET P =log a, loga^2/b , log(a^3/b^2),
If P is an arthimatical progression, then:
a2-a1 = a3-a2
=> loga^2/b - loga = log(a^3/b^2)- log (a^2/b)
We know that:
log a - log b = log a/b
==> log (a^2/b)/a = log (a^2/b)/(a^3/b^2)
==> log (a/b) = log (b/a)
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calendarEducator since 2010
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From in the information given:
a1 = loga
a2 = log(a^2 / b)
a3 = log(a^3/b^2)
Just from looking at it you can tell that it is, and the equation is log([a^n]/[b^(n-1)]); however, to solve it formulaically we do determine whether the difference between a1 and a2, and a2 and a3 is the same.
i.e. does a2 - a1 = a3 - a2
log(a^2/b) - loga = log(a^3/b^2) - log(a^2 / b
Using what we know about logs:
2loga - logb - loga = 3loga - 2logb - 2loga + logb
loga - logb = loga - logb
Therefore a1 - a2 = a2 - a3
This sequence is indeed an arithmetic series.
If T1 , T2 and T3 are the successive terms of an arithmetic progression (A P) then ,
T2 - T1 = T3 - T2 , the common diffrence.
Here T2 - T1 = log(a^2/b) - log a = log (a^2/b)(1/a) = log(a/b)........(1), as log x-logy = log (x+y).
T3 - T2 = log(a^2/b^2) = log (a^2/b) = log {(a^3/b^2)(b/a^2 ) }= log(a/b)...........(2).
So (1) and it is established that the common difference between the successive terms is a/b = T2-T1 = T3-T2. So the given 3 terms are in AP.
We'll have at least 2 methods to prove that.
We'll verify if the difference between 2 consecutive terms of the sequence is the same.
We'll note the consecutive terms as t1, t2, t3, where:
t1 = log a
t2 = log(a^2/b)
t3 = log(a^3/b^2)
We'll calculate the difference between t2 and t1:
t2 - t1 = loga - log(a^2/b)
We'll use the quotient property of the logarithms:
loga - log(a^2/b) = log (a*b/a^2)
We'll eliminate like terms:
log (a*b/a^2) = log (b/a)
t2 - t1 = log (b/a)
We'll calculate the difference between t3 and t2:
t3 - t2 = log(a^3/b^2) - log(a^2/b)
We'll use the quotient property of the logarithms, once again:
t3 - t2 = log (a^3 * b/b^2 * a^2)
We'll eliminate like terms:
t3 - t2 = log (a/b)
We notice that the difference between t2 and t1, t3 and t2 and so on is the same quantity: log (a/b).
So, the difference is the common difference between 2 consecutive terms of the sequence and the sequence is an Arithmetical Progression.
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