SOLUTION: The second term of a GP is 4, the fifth term is 81. Find the seventh term

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Question 1154077: The second term of a GP is 4, the fifth term is 81. Find the seventh term
Answer by MathLover1(20850) About Me  (Show Source):
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he second term of a GP is 4, the fifth term is 81. Find the seventh term
a%5B2%5D=4
a%5B5%5D=81

we need to find first term and common ratio
using nth term formula of a GP :

a%5Bn%5D+=+a%5B1%5D%2Ar%5E%28n-1%29 where a%5B1%5D is first term, r is common ratio, and n is a number of terms

given:

a%5B2%5D+=+a%5B1%5D%2Ar%5E%282-1%29
4=+a%5B1%5D%2Ar
r=4%2Fa%5B1%5D....eq.1

a%5B5%5D+=+a%5B1%5D%2Ar%5E%285-1%29
81=+a%5B1%5D%2Ar%5E4
r%5E4=81%2Fa%5B1%5D
r=root%284%2C81%2Fa%5B1%5D%29....eq.2


from eq.1 and eq.2 we have

4%2Fa%5B1%5D=root%284%2C81%2Fa%5B1%5D%29....take to fourth power

4%5E4%2Fa%5B1%5D%5E4=81%2Fa%5B1%5D

256%2Fa%5B1%5D%5E4=81%2Fa%5B1%5D

256=81a%5B1%5D%5E4%2Fa%5B1%5D

256=81a%5B1%5D%5E3

256%2F81=a%5B1%5D%5E3

a%5B1%5D%5E3=256%2F81

a%5B1%5D=root%283%2C256%2F81%29

a%5B1%5D=root%283%2C256%29%2Froot%283%2C81%29

a%5B1%5D=4root%283%2C4%29%2F3root%283%2C3%29

a%5B1%5D=%284%2F3%29%28root%283%2C4%29%2Froot%283%2C3%29%29

go to

r=4%2Fa%5B1%5D....eq.1, substitute a%5B1%5D

r=4%2F%284%2F3%29%28root%283%2C4%29%2Froot%283%2C3%29%29

r=12%2F%284%28root%283%2C4%29%2Froot%283%2C3%29%29%29

r=%283%2F2%29%2Aroot%283%2C+6%29




7th term: n=7





a%5B7%5D+=+1.46752322173%2A410.0625

a%5B7%5D+=601.8