SOLUTION: Determine whether the following goemetric series has a FINITE sum. If a finite sum exists, find it. 1. 8+4+2+.... 2. 2+3+9/2+.... Thanks!

Algebra ->  Sequences-and-series -> SOLUTION: Determine whether the following goemetric series has a FINITE sum. If a finite sum exists, find it. 1. 8+4+2+.... 2. 2+3+9/2+.... Thanks!      Log On


   

Question 84831: Determine whether the following goemetric series has a FINITE sum. If a finite sum exists, find it.
1. 8+4+2+....
2. 2+3+9/2+....
Thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
1.
It appears that this sequence is geometric, so to find the ratio between the two terms, simply divide any term you choose by the previous term. So lets pick 4, now divide that by 8
r=4%2F8=1%2F2
Now pick 2 and divide it by 4
r=2%2F4=1%2F2
So the ratio is 1%2F2 and our first term a%5B1%5D is 8. Since abs%28r%29%3C1 we can use the formula

S=a%2F%281-r%29 to find the finite sum

S=8%2F%281-1%2F2%29 Plug in a=8 and r=1%2F2

S=8%2F%281%2F2%29
S=%288%2F1%29%2A%282%2F1%29

S=16
So the finite sum is 16

If you want to verify this, take a look at the following pattern
Sum of the first 2 terms
8+4=12
Sum of the first 3 terms
8+4+2=14
Sum of the first 4 terms
8+4+2+1=15
Sum of the first 5 terms
8+4+2+1+0.5=15.5
Sum of the first 6 terms
8+4+2+1+0.5+0.25=15.75
Sum of the first 7 terms
8+4+2+1+0.5+0.25+0.125=15.875
Sum of the first 8 terms
8+4+2+1+0.5+0.25+0.125+0.0625=15.9375
Sum of the first 9 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125=15.96875
Sum of the first 10 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625=15.984375
Sum of the first 11 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125=15.9921875
Sum of the first 12 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625=15.99609375
Sum of the first 13 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125=15.998046875
Sum of the first 14 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625=15.9990234375
Sum of the first 15 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125=15.99951171875
Sum of the first 16 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125+0.000244140625=15.999755859375
Sum of the first 17 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125+0.000244140625+0.0001220703125=15.9998779296875
Sum of the first 18 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125+0.000244140625+0.0001220703125+6.103515625e-005=15.9999389648438
Sum of the first 19 terms
8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125+0.000244140625+0.0001220703125+6.103515625e-005+3.0517578125e-005=15.9999694824219
Sum of the first 20 terms

and you'll notice that the partial sums slowly approach 16. This verifies our answer.


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2.
Once again, it looks like this sequence is geometric. So lets find the ratio r:
r=3%2F2 Pick any term (I chose 3) and divide it by the previous term 2
r=%289%2F2%29%2F3=%289%2F2%29%281%2F3%29=9%2F6=3%2F2 Pick any term (I chose 9%2F2) and divide it by the previous term 3

So it appears that r=3%2F2 and our first term a=2. However, since r%3E1 this means our sum will not be finite. The reason why is because we keep adding on bigger and bigger numbers to our sum, which means it will grow to infinity.

Once again if you want to verify, take a look at this

Sum of the first 2 terms
1+1.5=2.5
Sum of the first 3 terms
1+1.5+2.25=4.75
Sum of the first 4 terms
1+1.5+2.25+3.375=8.125
Sum of the first 5 terms
1+1.5+2.25+3.375+5.0625=13.1875
Sum of the first 6 terms
1+1.5+2.25+3.375+5.0625+7.59375=20.78125
Sum of the first 7 terms
1+1.5+2.25+3.375+5.0625+7.59375+11.390625=32.171875
Sum of the first 8 terms
1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375=49.2578125
Sum of the first 9 terms
1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625=74.88671875
Sum of the first 10 terms
1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375=113.330078125
Sum of the first 11 terms
1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625=170.9951171875
Sum of the first 12 terms
1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625+86.49755859375=257.49267578125
Sum of the first 13 terms
1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625+86.49755859375+129.746337890625=387.239013671875
Sum of the first 14 terms
1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625+86.49755859375+129.746337890625+194.619506835938=581.858520507813
Sum of the first 15 terms
1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625+86.49755859375+129.746337890625+194.619506835938+291.929260253906=873.787780761719

and you can clearly see that the sums do not approach a finite number.