# Thread: Geometric and arithmetic progression

1. ## Geometric and arithmetic progression

a1, a2, a3 geometry progression

24/a1 , 60/a2 , 126/a3 arithmetic

the first 3 terms sum in geometric is 38 and the common ratio or quotient is less than 2

2. ## Re: Geometric and arithmetic progression

Originally Posted by Mustafashama
a1, a2, a3 geometry progression
24/a1 , 60/a2 , 126/a3 arithmetic
the first 3 terms sum in geometric is 38 and the common ratio or quotient is less than 2
$a_1=a,~a_2=a\cdot r,~\&~a_3=a\cdot r^2$ moreover $a+ar+ar^2=38$

3. ## Re: Geometric and arithmetic progression

You need a second equation to get a system of equations.

$\dfrac{126}{ar^2}-\dfrac{60}{ar}=\dfrac{60}{ar}-\dfrac{24}{a}$

Multiply out by $ar^2$ to get:
$126-60r=60r-24r^2$

$4r^2-20r+21=0$
$(4r-7)(r-3)=0$
Because the ratio is less than 2, this leaves only one solution. Plug that into the equation Plato showed you and solve for $a$.

4. ## Re: Geometric and arithmetic progression

Originally Posted by SlipEternal
You need a second equation to get a system of equations.

$\dfrac{126}{ar^2}-\dfrac{60}{ar}=\dfrac{60}{ar}-\dfrac{24}{a}$

Multiply out by $ar^2$ to get:
$126-60r=60r-24r^2$

$4r^2-20r+21=0$
$(4r-7)(r-3)=0$
Because the ratio is less than 2, this leaves only one solution. Plug that into the equation Plato showed you and solve for $a$.
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