SOLUTION: WHAT IS THE 30TH TERM OF GEOMETRIC SEQUENCE IF THE 10TH IS 3/512 AND 15TH IS 3/16384

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Question 949553: WHAT IS THE 30TH TERM OF GEOMETRIC SEQUENCE IF THE 10TH IS 3/512 AND 15TH IS 3/16384
Answer by Edwin McCravy(20056) About Me  (Show Source):
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WHAT IS THE 30TH TERM OF GEOMETRIC SEQUENCE IF THE 10TH IS 3/512 AND 15TH IS 3/16384
Use 

a%5Bn%5D=a%5B1%5Dr%5E%28n-1%29

Substitute n=10, a%5B10%5D=3%2F512=3%2F2%5E9

a%5B10%5D=a%5B1%5Dr%5E%2810-1%29
3%2F2%5E9=a%5B1%5Dr%5E9
Divide both sides by r%5E9
3%2F%282%5E9r%5E9%29=a%5B1%5D
3%2F%282r%29%5E9=a%5B1%5D

Substitute n=15, a%5B15%5D=3%2F16384=+3%2F2%5E14

a%5B15%5D=a%5B1%5Dr%5E%2815-1%29
3%2F2%5E14=a%5B1%5Dr%5E14
Divide both sides by r%5E14
3%2F%282%5E14r%5E14%29=a%5B1%5D
3%2F%282r%29%5E14=a%5B1%5D


Equate the two expressions for a%5B1%5D

3%2F%282r%29%5E9=3%2F%282r%29%5E14

Divide both sides by 3

1%2F%282r%29%5E9=1%2F%282r%29%5E14

Cross-multiply

%282r%29%5E14+=+%282r%29%5E9

Divide both sides by %282r%29%5E9

%282r%29%5E5+=+1

Take 5th roots of both sides

2r=1

r=1%2F2

Substitute 2r=1 in

3%2F%282r%29%5E9=a%5B1%5D

3%2F%281%29%5E9=a%5B1%5D

3%2F1=a%5B1%5D

3=a%5B1%5D

Now you finish by substituting n=30, a1=3, and r=1%2F2

into

a%5Bn%5D=a%5B1%5Dr%5E%28n-1%29

To find a30.

Edwin