Question 769801
Q:
If the difference between the 3rd and the 1st term of a G.P is 42, and the 4th term is greater than the 2nd term by 168, what is the (1) common ratio, (2) 1st term. (3) 4th term of the progression
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A:
{{{a[3] - a[1]}}} = 42 ----> {{{a[1]r^2 - a[1]}}} = 42 -----> {{{a[1]}}} = {{{42/(r^2-1)}}}  
{{{a[4]-a[2]}}} = 168 ----> {{{a[1]r^3-a[1]r}}} = 168 ----> {{{a[1]}}} = {{{168/(r^3-r)}}}
{{{42/(r^2-1)}}}  = {{{168/(r^3-r)}}}
{{{42r^3 - 42r}}} = {{{168r^2 - 168}}}
{{{42r^3 - 168r^2 - 42r + 168}}} = 0, divide by 42
{{{r^3 - 4r^2 - r + 4}}} = 0
(r - 1)(r + 1)(r - 4) = 0
r = 1, -1, or 4
Disregard r = 1 and -1 because it will make {{{a[1]}}} undefined.
common ratio, r ={{{highlight(4)}}}
1st term, {{{a[1]}}} = {{{42/(4^2-1)}}} = {{{highlight(14/5)}}}
4th term, {{{a[4]}}} = {{{a[1]r^3}}} = {{{(14/5)*(4^3)}}} = {{{highlight(896/5)}}}