Answer:
7
Explanation:
You want to know the number of terms in a geometric progression that has first term 16/3, last term 243/256, and a common ratio of 3/4.
Geometric Progression
The general term of a geometric progression is ...
an = a1(r^(n-1))
We want to find n when a1 = 16/3, an = 243/256, and r = 3/4.
Solution
243/256 = 16/3(3/4)^(n-1)
Multiplying by 3/16, we have ...
729/4096 = (3/4)^(n -1)
(3/4)^6 = (3/4)^(n-1) . . . . . . writing as powers of 3/4
6 = n -1 . . . . . . . . . . . . equating exponents
n = 7 . . . . . . . . . . add 1
The number of terms is 7.
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Additional comment
We can also use logarithms to find the value of n:
ln(729/4096) = (n -1)ln(3/4)
n = ln(729/4096)/ln(3/4) +1 = 7
The attachment shows this approach, along with the 7 terms of the sequence.
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