SOLUTION: Can someone help me with this? Thank you. infinite geometric series a+ar+ar^2+ar^3+... converges to 10, and the terms involving the even powers of r converges to 6. What is the v

Question 1112961: Can someone help me with this? Thank you.
infinite geometric series a+ar+ar^2+ar^3+... converges to 10, and the terms involving the even powers of r converges to 6. What is the value of a+r?
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
infinite geometric series a+ar+ar^2+ar^3+... can be written
:
summation from k = 0 to infinity of ar^k, since it converges we can write
:
1) a/(1-r) = 10
:
since the above series begins at k=0, I assume the series of even powers begins at k=0 also
:
summation from k = 0 to infinity of ar^(2k), since it converges we can write
:
2) a/(1-r^2) = 6
:
solve equation 1 for a
:
a = 10(1-r)
:
substitute for a in equation 2
:
10(1-r)/(1-r^2) = 6
:
(1-r)/(1-r)(1+r) = 6/10
:
1/(1+r) = 3/5
:
cross multiply the fractions
:
5 = 3 +3r
:
r = 2/3
:
a = 10(1-(2/3)) = 10/3
:
*************************
(10/3) + (2/3) = 12/3 = 4
*************************
:

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