Question 1085009
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(2) Let {a<sub>n</sub>} and {b<sub>n</sub>} for n = 1,2,3,... .

(i) The sequence {a<sub>n</sub>} satisfies the following relation:

                     {{{matrix(1,7,1[""]/a[n+1],""-"",1[""]/a[n],"",""="","",2)}}}.

Express the general term a<sub>n</sub> in terms of n when a<sub>1</sub> = {{{1/6}}}

Write down the the equation substituting n=1,2,3,4,...,n-1,n

{{{matrix(8,7,1[""]/a[2],""-"",1[""]/a[1],"",""="","",2,
              1[""]/a[3],""-"",1[""]/a[2],"",""="","",2,
              1[""]/a[4],""-"",1[""]/a[3],"",""="","",2,
              1[""]/a[5],""-"",1[""]/a[4],"",""="","",2,
              1[""]/a[2],""-"",1[""]/a[1],"",""="","",2,
              "...","","","","","","",
              1[""]/a[n],""-"",1[""]/a[n-1],"",""="","",2,
              1[""]/a[n+1],""-"",1[""]/a[n],"",""="","",2)}}}

Now add the equations and all the terms on the left cancel
except the first and last

{{{matrix(8,7,cross(1[""]/a[2]),""-"",1[""]/a[1],"",""="","",2,
              cross(1[""]/a[3]),""-"",cross(1[""]/a[2]),"",""="","",2,
              cross(1[""]/a[4]),""-"",cross(1[""]/a[3]),"",""="","",2,
              cross(1[""]/a[5]),""-"",cross(1[""]/a[4]),"",""="","",2,
              cross(1[""]/a[2]),""-"",cross(1[""]/a[1]),"",""="","",2,
              "...","","","","","","",
              cross(1[""]/a[n]),""-"",cross(1[""]/a[n-1]),"",""="","",2,
              1[""]/a[n+1],""-"",cross(1[""]/a[n]),"",""="","",2)}}}

So the sum of the left sides is the two terms that did not cancel,
and since there are n equations, the sum of the right sides is 2n. 

{{{matrix(1,7,-1[""]/a[1],""+"",1[""]/a[n+1],"",""="","",2n)}}}

{{{matrix(1,7,-1[""]/a[1],""+"",1[""]/a[n+1],"",""="","",2n)}}}

Since {{{a[1]=1/6}}}, {{{-1^""/a[1]=-6

{{{matrix(1,7,-6,""+"",1[""]/a[n+1],"",""="","",2n)}}}

{{{matrix(1,5,1[""]/a[n+1],"",""="","",2n+6)}}}

Take reciprocals of both sides:

{{{matrix(1,5,a[n+1],"",""="","",1/(2n+6))}}}

Let n = m-1

{{{matrix(1,5,a[(m-1)+1],"",""="","",1/(2(m-1)+6))}}}
{{{matrix(1,5,a[m-1+1],"",""="","",1/(2m-2+6))}}}
{{{matrix(1,5,a[m],"",""="","",1/(2m+4))}}}

Let m = n

{{{matrix(1,5,a[n],"",""="","",1/(2n+4))}}}

-----------------------------------------------------
Edwin</pre>