SOLUTION: a) Find the sum of this sequence: 1+3+9+27..3^19 b) Form the formula by filling in the blank: S= a + ar + ar^2 + ar^3 + ... + ar^(n-1)(The series stops at n-1 power because there

Algebra ->  Algebra  -> Sequences-and-series -> SOLUTION: a) Find the sum of this sequence: 1+3+9+27..3^19 b) Form the formula by filling in the blank: S= a + ar + ar^2 + ar^3 + ... + ar^(n-1)(The series stops at n-1 power because there      Log On

Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations!
Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!

   

Question 241243: a) Find the sum of this sequence: 1+3+9+27..3^19
b) Form the formula by filling in the blank:
S= a + ar + ar^2 + ar^3 + ... + ar^(n-1)(The series stops at n-1 power because there are n terms, but the first one is not multiplied by r)
rS=______________?
rS=___________?(Substritute S-a)
rS-S=______?
but rS-S can be factored: rS-S=S(r-1),so
S=__________?


I got a), it is 1,743,392,200 But i have no idea how to come up with the formula, even though the book reads the general formula is S=a(r^n-1)/(r-1)
Please explain. I'm very thankful for your helps!

Answer by rapaljer(4551) About Me  (Show Source):
You can put this solution on YOUR website!
S= a + ar + ar^2 + ar^3 + ... + ar^(n-1)(The series stops at n-1 power because there are n terms, but the first one is not multiplied by r)
rS=ar +ar^2 + ar^3 + . . . + ar^n

Subtract the above equations. All the middle terms subtract out!
S-rS = a - ar^n

Factor out the S on the left and the a on the right side of the equation:
S(1-r) = a(1-r^n)

Divide both sides by (1-r):
S=%28a%281-r%5En%29%29%2F%281-r%29

This does not agree exactly with your final answer, so multiply both numerator and denominator by -1, and it will give you the final result you are looking for!

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus