Question 175764
Find all the geometric sequences such that the sum of the first two terms is 2 and the sum of the first three terms is 3. 
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You have this system of equations:

{{{system(a[1] + a[2] = 2,
a[1] + a[2] + a[3] = 3)}}}

Now if you subtract the first equations
term by term from the second equation
you get

{{{a[3] = 1}}}

{{{a[1]r^(3-1) = 1}}}

{{{a[1]r^2 = 1}}}

{{{a[1] = 1/r^2}}}

{{{a[2] = a[1]r = (1/r^2)r = (1/r^cross(2))cross(r) = 1/r}}}

So the first three terms are

{{{a[1] = 1/r^2}}}
{{{a[2] = 1/r}}}
{{{a[3] = 1}}}

and {{{a[n]}}} is given by:
 
{{{a[n] = a[1]r^(n-1) = (1/r^2)(r^(n-1)) = r^(n-3)}}}

So all such geometric sequences are of the form:

{{{1/r^2}}}, {{{1/r}}}, {{{1}}}, {{{r}}}, {{{r^2}}}, {{{r^3}}},···,{{{r^(n-3)}}},···

Edwin</pre>