Question 1170436
.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;There are &nbsp;<U>two different progressions</U> &nbsp;and &nbsp;<U>two different answers</U>.



The common ratio is  &nbsp;&nbsp;+/- {{{sqrt(9)}}} = +/- 3.



<U>Case 1.  &nbsp;&nbsp;Common ratio is &nbsp;3</U>


<pre>
            Then 


            a1 is unknown
            a2 = 3a1
            a3 = 9a1
            a5 = 81a1
            a6 = 243a1
            a7 = 729a1
            the last two terms add to 243a1 + 729a1 = 972a1 = 1944
            so a1 = 2, and r = 3
            -
            sum of first 10 terms is   2*(1-3^10)/(-2) = 59048
            sum of first  3 terms is   2*(1-3^3)/(-2)  =    26
            the sum of the 4th to 10th terms inclusive is  59048  - 26 = 59022
            the series is 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366.
</pre>


<U>Case 2.  &nbsp;&nbsp;Common ratio is &nbsp;-3</U>


<pre>
            Then 


            a1 is unknown
            a2 =   -3a1
            a3 =    9a1
            a5 =  -81a1
            a6 =  243a1
            a7 = -729a1
            the last two terms add to -729a1 + 243a1 = -486a1 = 1944
            so a1 = -4, and r = -3
            -
            sum of first 10 terms is   (-4)*(1-(-3)^10)/4 = 59048
            sum of first  3 terms is   (-4)*(1-(-3)^3)/4  =   -28
            the sum of the 4th to 10th terms inclusive is  59048 - (-28) = 59076
            the series is -4, 12, -36, 108, -324, 972, -2916, , 8748, -26244.
</pre>

Solved.



Do not accept any other answer/answers.