Question 882717
Consider the sequence x+3, x+7, 4x-2...
<pre>
a) if the sequence is arithmetic, then 

{{{(matrix(2,1,second, term))}}}{{{""-""}}}{{{(matrix(2,1,first,term))}}} {{{""=""}}} {{{(matrix(2,1,third, term))}}}{{{""-""}}}{{{(matrix(2,1,second, term))}}} {{{""=""}}} {{{(matrix(2,1,common, difference))}}}

     {{{matrix(1,9,(x+7),""-"",(x+3),"",""="","",(4x-2),""-"",(x+7))}}}
          {{{x+7-x-3}}}{{{""=""}}}{{{4x-2-x-7}}}
                 {{{4}}}{{{""=""}}}{{{3x-9}}}      <-- so common difference = 4
                {{{13}}}{{{""=""}}}{{{3x}}}
                {{{13/3}}}{{{""=""}}}{{{x}}}

first term = x+3 = {{{13/3+3}}} = {{{13/3+9/3}}} = {{{22/3}}}
second term = {{{22/3+4}}} = {{{22/3+12/3}}} = {{{34/3}}}
As a check,
second term = x+7 = {{{13/3+7}}} = {{{13/3+21/3}}} = {{{34/3}}}
third term = {{{34/3+4}}} = {{{34/3+12/3}}} = {{{46/3}}}
As a check,
third term = 4x-2 = {{{4(13/3)-2}}} = {{{(52/3)-6/3}}}= {{{46/3}}}
fourth term = {{{46/3+4}}} = {{{46/3+12/3}}} = {{{58/3}}}


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b) if the sequence is geometric, then 

{{{(matrix(2,1,second, term))}}}{{{"÷"}}}{{{(matrix(2,1,first,term))}}} {{{""=""}}} {{{(matrix(2,1,third, term))}}}{{{"÷"}}}{{{(matrix(2,1,second, term))}}} {{{""=""}}} {{{(matrix(2,1,common, ratio))}}}

            {{{matrix(1,5,(x+7)/(x+3),"",""="","",(4x-2)/(x+7))}}}

        {{{matrix(1,5,(x+7)(x+7),"",""="","",(4x-2)(x+3))}}}
              
        {{{matrix(1,5,x^2+14x+49,"",""="","",4x^2+10x-6)}}}

        {{{matrix(1,5,-3x^2+4x+55,"",""="","",0)}}}

        {{{matrix(1,5,3x^2-4x-55,"",""="","",0)}}}

        {{{matrix(1,5,(x-5)(3x+11),"",""="","",0)}}}

Two solutions:
       
    {{{matrix(1,5,x-5,"",""="","",0)}}}, {{{matrix(1,5,3x+11,"",""="","",0)}}}

      {{{matrix(1,5,x,"",""="","",5)}}},    {{{matrix(1,5,3x,"",""="","",-11)}}}

                    {{{matrix(1,5,x,"",""="","",-11/3)}}}

Using x=5

first term = x+3 = {{{5+3}}} = {{{8}}}
second term = x+7 = {{{5+7}}} = {{{12}}}
common ratio = {{{matrix(1,2,second,term)/matrix(1,2,first,term)}}} = {{{12/8}}} = {{{3/2}}}
third term = {{{12*expr(3/2)}}} = {{{18}}}
As a check,
third term = 4x-2 = {{{4(5)-2}}} = {{{20expr-2}}} = {{{18}}}
fourth term = {{{18*expr(3/2)}}} = {{{27}}}

Using x={{{-11/3}}}

first term = x+3 = {{{-11/3+3}}} = {{{-11/3+9/3}}} = {{{-2/3}}}
second term = x+7 = {{{-11/3+7}}} = {{{-11/3+21/3}}} = {{{10/3}}}
common ratio = {{{matrix(1,2,second,term)/matrix(1,2,first,term)}}} = {{{((10/3))/((-2/3))}}} = {{{(10/3)(-3/2)}}}{{{-5}}}
third term = {{{(10/3)(-5)}}} = {{{-50/3}}}
As a check,
third term = 4x-2 = {{{4(-11/3)-2}}} = {{{(-44/3)-6/3}}}= {{{-50/3}}}
fourth term = {{{(-50/3)(-5)}}} = {{{250/3}}}

Edwin</pre>