Question 84831
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=4/8=1/2}}}

Now pick 2 and divide it by 4

{{{r=2/4=1/2}}}

So the ratio is {{{1/2}}} and our first term {{{a[1]}}} is 8. Since {{{abs(r)<1}}} we can use the formula 


{{{S=a/(1-r)}}} to find the finite sum


{{{S=8/(1-1/2)}}} Plug in a=8 and {{{r=1/2}}}


{{{S=8/(1/2)}}}

{{{S=(8/1)*(2/1)}}}


{{{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. 



-------------------------------------------------------------


2.

Once again, it looks like this sequence is geometric. So lets find the ratio r:

{{{r=3/2}}} Pick any term (I chose 3) and divide it by the previous term 2

{{{r=(9/2)/3=(9/2)(1/3)=9/6=3/2}}} Pick any term (I chose {{{9/2}}}) and divide it by the previous term 3


So it appears that {{{r=3/2}}} and our first term {{{a=2}}}. However, since {{{r>1}}} 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.