Question 773910
Q:
the sum of first 4 terms of Geometric progression is 7.5 if the sum of middle two terms is 3 find the terms of gp
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A:
Let x = first term
and r = common ratio
Therefore, rx = second term
           {{{r^2x}}} = third term
           {{{r^3x}}} = fourth term
{{{x + rx + r^2x + r^3x}}} = 7.5
{{{x(1 + r) + r^2x(1 + r)}}} = 7.5
{{{(1 + r)(x + r^2x)}}} = 7.5
{{{x(1 + r)(1 + r^2)}}} = 7.5
The sum of the middle two terms is
{{{rx + r^2x}}} = 3  ---> rx(1 + r) = 3   ---> x = {{{3/r(1+r)}}}
{{{(3/r(1+r))(1 + r)(1 + r^2)}}} = 7.5
{{{(3/r(cross(1+r)))(cross(1 + r))(1 + r^2)}}} = 7.5
{{{1 + r^2}}} = {{{7.5(r/3)}}}
{{{r^2 - 2.5r + 1 = 0}}}
{{{2r^2 - 5r + 2 = 0}}}
(2r - 1)(r - 1) = 0
r = {{{1/2}}} or r = 1, disregard r = 1
If r = 1/2, then x = {{{3/(1/2)(1+(1/2))}}} = 4
The first term is 4, to get the next term, multiply by the common ratio 1/2.
The first 4 terms are {{{highlight(4)}}}, {{{highlight(2)}}}, {{{highlight(1)}}}, and {{{highlight(1/2)}}}.