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odolwa99
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Homework Statement
Q.: Show that if log a, log b and log c are three consecutive terms of an arithmetic sequence, then a, b and c are in geomtric sequence.
Homework Equations
Un = a + (n - 1)d and Sn = [itex]\frac{a(r^n - 1)}{r - 1}[/itex]
The Attempt at a Solution
Attempt:
Consider arithmetic sequence log 2, log 4, log 8 or log 3, log 9, log 27
The respective values of each sequence is 0.301, 0.602, 0.903 and 0.477, 0.954, 1.431
The difference between the numbers in the first sequence is 0.301 and 0.477 in the second.
Geoetric sequence is a, ar, ar^2. So it follows from the above sequnces that...
a = 2 or 3
ar = 2.2 = 4 or 3.3 = 9
ar^2 = 2.2^2 = 8 or 3.3^2 = 27
Thus a, b and c can be expressed in arithmetic sequence as log values and then as geometric sequences without log.
Answer: From textbook: b^2 = ac [itex]\Rightarrow[/itex] Geometric sequence.
I feel I've adequately proven the point, albeit in a slightly unconventional way. Can someone steer me toward the book's answer, or is my method already satisfactory? Thank you.