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A geometric sequence has first term 54 and 4th term 2. What is the common ratio?

Answer 1

Serenity Jones

#r = 1/3#

#A_n = A_1r^(n - 1)#

#A_1 = 54#

#A_4 = 2 = A_1r^(4 - 1)#

#=> 2 = 54r^3#

#=> 2/54 = r^3# #=> 1/27 = r^3#

#=> (1/3)^3 = r^3#

#=> r = 1/3#

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Answer 2

Elias Ackerman

To find the common ratio ( r ) of a geometric sequence, we can use the formula ( a_n = a_1 \times r^{(n-1)} ), where ( a_n ) is the ( n )-th term, ( a_1 ) is the first term, and ( r ) is the common ratio.

Given that the first term ( a_1 = 54 ) and the fourth term ( a_4 = 2 ), we have:

[ a_4 = a_1 \times r^{(4-1)} ] [ 2 = 54 \times r^3 ]

Now, solving for ( r ):

[ r^3 = \frac{2}{54} ] [ r^3 = \frac{1}{27} ]

Taking the cube root of both sides:

[ r = \sqrt[3]{\frac{1}{27}} ] [ r = \frac{1}{3} ]

So, the common ratio of the geometric sequence is ( \frac{1}{3} ).

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Answer from HIX Tutor

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