Arithmetic Progression Question: I have no numbers and I am not sure how to proceed
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Three numbers are consecutive terms of a geometric progression. If we add 2 to the second number, the
new progression becomes arithmetic. If we add 9 to the third number, the progression becomes geometric.
Find the original numbers.
sequences-and-series arithmetic
asked Jan 6 at 18:53
MzeroMzero
31
$endgroup$
add a comment |
$begingroup$
Three numbers are consecutive terms of a geometric progression. If we add 2 to the second number, the
new progression becomes arithmetic. If we add 9 to the third number, the progression becomes geometric.
Find the original numbers.
sequences-and-series arithmetic
asked Jan 6 at 18:53
MzeroMzero
31
$endgroup$
$begingroup$
Start by writing down what you do know. The three numbers are $a,ar,ar^2$ Now use the second and third sentences to write equations. Presumably the third sentence still has the $2$ added to the second number before you add the $9$. You will have two equations in the two unknowns $a,r$
$endgroup$
– Ross Millikan
Jan 6 at 19:17
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@RossMillikan I did exactly that but this is where I get stuck. What common thing do these three equations have when I have no numbers? a+ar+ar^2; a+ar+ar^2+2; a+ar+ar^2+11
$endgroup$
– Mzero
Jan 6 at 20:02
add a comment |
$begingroup$
Three numbers are consecutive terms of a geometric progression. If we add 2 to the second number, the
new progression becomes arithmetic. If we add 9 to the third number, the progression becomes geometric.
Find the original numbers.
sequences-and-series arithmetic
asked Jan 6 at 18:53
MzeroMzero
31
$endgroup$
Three numbers are consecutive terms of a geometric progression. If we add 2 to the second number, the
new progression becomes arithmetic. If we add 9 to the third number, the progression becomes geometric.
Find the original numbers.
sequences-and-series arithmetic
sequences-and-series arithmetic
asked Jan 6 at 18:53
MzeroMzero
31
asked Jan 6 at 18:53
MzeroMzero
31
asked Jan 6 at 18:53
MzeroMzero
31
asked Jan 6 at 18:53
MzeroMzero
31
asked Jan 6 at 18:53
MzeroMzero
31
31
$begingroup$
Start by writing down what you do know. The three numbers are $a,ar,ar^2$ Now use the second and third sentences to write equations. Presumably the third sentence still has the $2$ added to the second number before you add the $9$. You will have two equations in the two unknowns $a,r$
$endgroup$
– Ross Millikan
Jan 6 at 19:17
$begingroup$
@RossMillikan I did exactly that but this is where I get stuck. What common thing do these three equations have when I have no numbers? a+ar+ar^2; a+ar+ar^2+2; a+ar+ar^2+11
$endgroup$
– Mzero
Jan 6 at 20:02
add a comment |
$begingroup$
Start by writing down what you do know. The three numbers are $a,ar,ar^2$ Now use the second and third sentences to write equations. Presumably the third sentence still has the $2$ added to the second number before you add the $9$. You will have two equations in the two unknowns $a,r$
$endgroup$
– Ross Millikan
Jan 6 at 19:17
$begingroup$
@RossMillikan I did exactly that but this is where I get stuck. What common thing do these three equations have when I have no numbers? a+ar+ar^2; a+ar+ar^2+2; a+ar+ar^2+11
$endgroup$
– Mzero
Jan 6 at 20:02
$begingroup$
Start by writing down what you do know. The three numbers are $a,ar,ar^2$ Now use the second and third sentences to write equations. Presumably the third sentence still has the $2$ added to the second number before you add the $9$. You will have two equations in the two unknowns $a,r$
$endgroup$
– Ross Millikan
Jan 6 at 19:17
$begingroup$
Start by writing down what you do know. The three numbers are $a,ar,ar^2$ Now use the second and third sentences to write equations. Presumably the third sentence still has the $2$ added to the second number before you add the $9$. You will have two equations in the two unknowns $a,r$
$endgroup$
– Ross Millikan
Jan 6 at 19:17
$begingroup$
@RossMillikan I did exactly that but this is where I get stuck. What common thing do these three equations have when I have no numbers? a+ar+ar^2; a+ar+ar^2+2; a+ar+ar^2+11
$endgroup$
– Mzero
Jan 6 at 20:02
$begingroup$
@RossMillikan I did exactly that but this is where I get stuck. What common thing do these three equations have when I have no numbers? a+ar+ar^2; a+ar+ar^2+2; a+ar+ar^2+11
$endgroup$
– Mzero
Jan 6 at 20:02
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
To have an arithmetic progression, you need a common difference.
The second talks of adding $2$ to the second number, so $a,ar+2,ar^2$ is an arithmetic progression, so $ar^2-(ar+2)=(ar+2)-a$.
Then you get back to geometric by adding $9$ to the third, so $frac {ar+2}a=frac {ar^2+9}{ar+2}$.
There are your two equations.
answered Jan 6 at 20:41
Ross MillikanRoss Millikan
300k24200375
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
To have an arithmetic progression, you need a common difference.
The second talks of adding $2$ to the second number, so $a,ar+2,ar^2$ is an arithmetic progression, so $ar^2-(ar+2)=(ar+2)-a$.
Then you get back to geometric by adding $9$ to the third, so $frac {ar+2}a=frac {ar^2+9}{ar+2}$.
There are your two equations.
answered Jan 6 at 20:41
Ross MillikanRoss Millikan
300k24200375
$endgroup$
add a comment |
$begingroup$
To have an arithmetic progression, you need a common difference.
The second talks of adding $2$ to the second number, so $a,ar+2,ar^2$ is an arithmetic progression, so $ar^2-(ar+2)=(ar+2)-a$.
Then you get back to geometric by adding $9$ to the third, so $frac {ar+2}a=frac {ar^2+9}{ar+2}$.
There are your two equations.
answered Jan 6 at 20:41
Ross MillikanRoss Millikan
300k24200375
$endgroup$
add a comment |
$begingroup$
To have an arithmetic progression, you need a common difference.
The second talks of adding $2$ to the second number, so $a,ar+2,ar^2$ is an arithmetic progression, so $ar^2-(ar+2)=(ar+2)-a$.
Then you get back to geometric by adding $9$ to the third, so $frac {ar+2}a=frac {ar^2+9}{ar+2}$.
There are your two equations.
answered Jan 6 at 20:41
Ross MillikanRoss Millikan
300k24200375
$endgroup$
To have an arithmetic progression, you need a common difference.
The second talks of adding $2$ to the second number, so $a,ar+2,ar^2$ is an arithmetic progression, so $ar^2-(ar+2)=(ar+2)-a$.
Then you get back to geometric by adding $9$ to the third, so $frac {ar+2}a=frac {ar^2+9}{ar+2}$.
There are your two equations.
answered Jan 6 at 20:41
Ross MillikanRoss Millikan
300k24200375
answered Jan 6 at 20:41
Ross MillikanRoss Millikan
300k24200375
answered Jan 6 at 20:41
Ross MillikanRoss Millikan
300k24200375
answered Jan 6 at 20:41
Ross MillikanRoss Millikan
300k24200375
300k24200375
add a comment |
add a comment |
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$begingroup$
Start by writing down what you do know. The three numbers are $a,ar,ar^2$ Now use the second and third sentences to write equations. Presumably the third sentence still has the $2$ added to the second number before you add the $9$. You will have two equations in the two unknowns $a,r$
$endgroup$
– Ross Millikan
Jan 6 at 19:17
$begingroup$
@RossMillikan I did exactly that but this is where I get stuck. What common thing do these three equations have when I have no numbers? a+ar+ar^2; a+ar+ar^2+2; a+ar+ar^2+11
$endgroup$
– Mzero
Jan 6 at 20:02