SOLUTION: the second term of a geometric progression is 36 more than the first term. the difference between the fourth and the third is 900. calculate the common ratio and the first term.

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Question 961086: the second term of a geometric progression is 36 more than the first term. the difference between the fourth and the third is 900. calculate the common ratio and the first term.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
the second term of a geometric progression 
a%5B2%5D=a%5B1%5Dr

is 36 more than the first term.
a%5B2%5D=a%5B1%5D%2B36
a%5B1%5Dr=a%5B1%5D%2B36
a%5B1%5Dr-a%5B1%5D=36
a%5B1%5D%28r-1%29=36
a%5B1%5D=36%2F%28r-1%29

the difference between the fourth and the third is 900.
a%5B4%5D-a%5B3%5D=900
a%5B4%5D=a%5B1%5Dr%5E%284-1%29=a%5B1%5Dr%5E3, a%5B3%5D=a%5B1%5Dr%5E%283-1%29=a%5B1%5Dr%5E2
a%5B1%5Dr%5E3-a%5B1%5Dr%5E2=900
a%5B1%5Dr%5E2%28r-1%29=900

Substitute a%5B1%5D=36%2F%28r-1%29

%2836%2F%28r-1%29%29r%5E2%28r-1%29=900

%2836%2Fcross%28r-1%29%29r%5E2%28cross%28r-1%29%29=900

36r%5E2=900

r%5E2=25

r+=+%22%22+%2B-+5

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Using the positive value of r:

a%5B1%5D=36%2F%28r-1%29
a%5B1%5D=36%2F%285-1%29
a%5B1%5D=36%2F4
a%5B1%5D=9

One solution r=5, a1=9

Checking:
Sequence is 9,45,225,1125
2nd term 45 is 36 more than 1st term 9.  9+36=45
Difference between 4th and 3rd term is 1125-225=900 

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Using the negative value of r:

a%5B1%5D=36%2F%28r-1%29
a%5B1%5D=36%2F%28-5-1%29
a%5B1%5D=36%2F%28-6%29
a%5B1%5D=-6

Second solution r=-5, a1=-6

Checking:
Sequence is -6,30,-150,750
2nd term 30 is 36 more than 1st term -6.  -6+36=30
Difference between 4th and 3rd term is 750-(-150)=750+150=900

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Two solutions.

Edwin