The sum of three consecutive terms of a geometric sequence is 73.5. If the product of these terms is 2744, what are the terms?

Question

anonymous · Answer

56 , 14 and 3.5 
56+14+3.5=73.5
56*14*3.5=2744

anonymous · Answer

How did you get that?

anonymous · Answer

this question was asked during one of my mathematical examinations..i still remember it as a matter of fact

anonymous · Answer

WOAH. That is awesome. SO how did you solve for it when you were taking the exam?

anonymous · Answer

Let x be the first element and r be the common ratio
x + r x + r^2 x = 73.5
x (rx)(r^2x)= 2744
You solve the two equations for x and r
and you find two real solutions
x=3.5 and r =4
or
x=56 and r =1/4
Both yield the same 4 numbers.

anonymous · Answer

how is it r^2?

anonymous · Answer

let us suppose  the three terms are $${{a}\over{r}}, {a}, {ar}$$
then
$${{a}\over{r}}+{a}+{ar} = 73.5$$
 
and $${{a}\over{r}}\times{a}\times{ar}=2744$$
$${a^3}=2744$$
$${a}=14$$
now plug in this value in first equation
$${a}({{1}\over{r}}+{1}+{r}) = 73.5$$
$${14}({{1+r+r^2}\over{r}}) = 73.5$$
$${14}({{1+r+r^2}}) = 73.5r$$
$${14 +14r + 14r^2} = 73.5r$$
$${14r^2 -59.5r + 14} = 0$$
now solve for r

anonymous · Answer

.0O0. I bow down to your awesomeness! Thank you!!!!!

anonymous · Answer

still having any difficulty ?

anonymous · Answer

well, why do you assume that they are a/r, a, and ar?

anonymous · Answer

because they are three consecutive numbers of the GP each having a common ratio in this case Common ration is r

anonymous · Answer

and i was having multiplication so r gets cancelled and easy for calculation

anonymous · Answer

GP?

anonymous · Answer

Geometric Sequence or, Geometric Progression or, Geometric Series

anonymous · Answer

oh! so like that other person was saying about the a+ar+ar^2 and stuff?

anonymous · Answer

yeah both mean the same

anonymous · Answer

SO I don't get how you are supposed to do this... I started factoring out things, but it's confusing...idk how to do...

anonymous · Answer

Let x be the first element and r be the common ratio
 x + r x + r^2 x = 73.5 
x (rx)(r^2x)= 2744 
You solve the two equations for x and r and you find two real solutions x=3.5 and r =4 or x=56 and r =1/4 Both yield the same 4 numbers.

Notice that the second equation yields
r x =14  so x =14/r
14/r + 14 + 14 r = 73.5
Which yields r=4 or r = 1/4

anonymous · Answer

uh...

anonymous · Answer

okay.....