How to Solve for the Corresponding Sequences

Question

Consider the following question:

Three numbers are in arithmetic progression. Three other numbers are in geometric progression. Adding the corresponding terms of these progressions we obtain: 22, 50, 105. The sum of the numbers in the arithmetic progression is 168. Find the two progressions.

Based on this information, we can suppose the arithmetic sequence is $x, x+d, x+2d$ and the geometric sequence is $y, ay, a^2y$. Using this notation we can identify the following equations:

$x + y = 22$

$x + d + ay = 105$

$x + 2d + a^2y = 105$

$3x + 3d = 168$

I've tried playing around with these equations a bit with some repeated substitution and adding or subtracting the given equations but I haven't made a ton of progress. The only progress has been that $y = \frac{-6}{a}$, $y(a-1)^2 = 27$ and $x + d = 56$.

Could anyone please provide some guidance on what steps should be followed to be able to solve for the individual variables? Perhaps there's a different approach I'm not seeing.

Answer 1

Notice that the first equation you have posted sould be $x+y=22$ and the second should be: $x+d+ay=50$. From the first and the second you have found it is immediate to find: $$ ya(a-1)^2=27a\Leftrightarrow (a-1)^2=-\frac{9}{2}a $$ hence $a=-2$ or $a=-\frac{1}{2}$ and $y=3$ or $y=12$. Then $x=19$ or $x=10$ and at last from the last $d=37$ or $d=46$

Answer 2

Just substitute $y = \frac{-6}{a}$ in $y(a-1)^2 = 27$ and solve the quadratic equation $$-6(a-1)^2 = 27a.$$ From this you get $a$, then $y$, then $x$, then $d$.