SOLUTION: An infinite geo series has the sum 16. If the sum of the first two terms is 15, find a1 and r

Algebra ->  Sequences-and-series -> SOLUTION: An infinite geo series has the sum 16. If the sum of the first two terms is 15, find a1 and r       Log On


   



Question 849306: An infinite geo series has the sum 16. If the sum of the first two terms is 15, find a1 and r

Answer by AnlytcPhil(1807) About Me  (Show Source):
You can put this solution on YOUR website!
An infinite geo series has the sum 16.
S%5Binfinity%5D%22%22=%22%22a%5B1%5D%2F%281-r%29

a%5B1%5D%2F%281-r%29%22%22=%22%2216

a%5B1%5D%22%22=%22%2216%281-r%29

the sum of the first two terms is 15
a%5B2%5D%22%22%2B%22%22a%5B1%5D%22%22=%22%2215

a%5B2%5D%22%22%2B%22%22a%5B1%5D%2Ar

a%5B1%5D%2Ar%22%22%2B%22%22a%5B1%5D%22%22=%22%2215

a%5B1%5D%28r%2B1%29%22%22=%22%2215

So we have the system:

system%28a%5B1%5D=16%281-r%29%2C+a%5B1%5D%28r%2B1%29=15%29

Use substitution:

16%281-r%29%28r%2B1%29%22%22=%22%2215

16%28r%2B1-r%5E2-r%29%22%22=%22%2215

16%281-r%5E2%29%22%22=%22%2215 

16-16r%5E2%22%22=%22%2215

-16r%5E2%22%22=%22%22-1

r%5E2%22%22=%22%221%2F16

r%22%22=%22%22%22%22+%2B-+1%2F4

using r=1%2F4

a%5B1%5D%22%22=%22%2216%281-1%2F4%29
a%5B1%5D%22%22=%22%2216%283%2F4%29
a%5B1%5D%22%22=%22%2212

using r=-1%2F4

a%5B1%5D%22%22=%22%2216%281-%28-1%2F4%29%29
a%5B1%5D%22%22=%22%2216%281%2B1%2F4%29
a%5B1%5D%22%22=%22%2216%285%2F4%29
a%5B1%5D%22%22=%22%2220

Two solutions:

a%5B1%5D%22%22=%22%2212, r=1%2F4

and

a%5B1%5D%22%22=%22%2220, r=-1%2F4
 
Edwin