Question 1209818
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The AI solution from the other "tutor" has the right final answer, but the AI presentation is hard to follow.  Here is a solution in English....<br>
{{{a(0)=1}}}
{{{a(2)=2}}}
{{{a(n+2)=a(n+1)/a(n)}}}<br>
Although the given starting information is not standard, we can use the recursive definition with the given values of {{{a(0)}}} and {{{a(2)}}} to find {{{a(1)}}} and then use the recursive definition to find all subsequent terms.<br>
For n=0....<br>
{{{a(2)=a(1)/a(0)}}}
{{{2=a(1)/1}}}
{{{a(1)=2}}}<br>
So the first three terms are 1, 2, and 2.<br>
Note that the first term is {{{a(0)}}} and the problem asks for the sum of the terms through {{{a(100)}}}, so we are looking for the sum of the first 101 terms.<br>
{{{a(3)=a(2)/a(1)=2/2=1}}}<br>
{{{a(4)=a(3)/a(2)=1/2=1/2}}}<br>
{{{a(5)=a(4)/a(3)=(1/2)/1=1/2}}}<br>
{{{a(6)=a(5)/a(4)=(1/2)/(1/2)=1}}}<br>
{{{a(7)=a(6)/a(5)=1/(1/2)=2}}}<br>
{{{a(8)=a(7)/a(6)=2/1=1}}}<br>
We now see that the terms of the sequence follow a cyclic pattern of length 6:<br>
1, 2, 2, 1, 1/2, 1/2<br>
With a cycle of length 6, 101 terms will mean 16 times through the cycle plus 5 more terms.<br>
The sum of the numbers in each cycle is 7; the sum of the numbers in 16 cycles is 16*7 = 112.
The sum of the first 5 numbers in each cycle is 6.5.
The sum of the first 101 terms is 112+6.5 = 118.5<br>
ANSWER: 118.5<br>