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If a, b, c are in G.P, prove that the following are also in $G.P.:{{a}^{3}},{{b}^{3}},{{c}^{3}}$

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Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: Try to find the relation between a, b, c using geometric progression definition and then try to use it in the series of ${{a}^{3}},{{b}^{3}},{{c}^{3}}$ and prove that it is in GP.

In the question we are given that a, b, c are in G.P.
The above statement means that there exists a number ‘r’ such that ‘a’ when multiplied with ‘r’ gives ‘b’ and ‘b’ when multiplied with ‘r’ gives ‘c’.
Then we can represent this above relation as,
$b=ar$ and $c=br$
So, $c=br=\left( ar \right)\times r=a{{r}^{2}}$
Hence, the numbers $a,b,c$ can be represented as $a,ar,a{{r}^{2}}$.
Now, we are asked about ${{a}^{3}},{{b}^{3}},{{c}^{3}}$.
So, we can transform ${{a}^{3}},{{b}^{3}},{{c}^{3}}$ as ${{a}^{3}},{{a}^{3}}{{r}^{3}},{{a}^{3}}{{r}^{6}}$.
Here, in the ${{a}^{3}},{{a}^{3}}{{r}^{3}},{{a}^{3}}{{r}^{6}}$ we see that there exists a new number ${{r}^{3}}$ which multiplied to ${{a}^{3}}$ become ${{a}^{3}}{{r}^{3}}$ which is ${{b}^{3}}$, which when further multiplied becomes ${{a}^{3}}{{r}^{6}}$ which is ${{c}^{3}}$.
Geometric progression series is a series in which the ratio between two consecutive series is the same.
So we will check here, by substituting the values,
$\dfrac{{{b}^{3}}}{{{a}^{3}}}=\dfrac{{{(ar)}^{3}}}{{{a}^{3}}}$
Cancelling the like terms, we get the ratio as
$\dfrac{{{b}^{3}}}{{{a}^{3}}}={{r}^{3}}.........(i)$
Similarly,
$\dfrac{{{c}^{3}}}{{{b}^{3}}}=\dfrac{{{(a{{r}^{2}})}^{3}}}{{{\left( ar \right)}^{3}}}$
Cancelling the like terms, we get the ratio as
$\dfrac{{{c}^{3}}}{{{b}^{3}}}={{r}^{3}}.........(ii)$
Comparing equations (i) and (ii), we see that the ratios are the same.
Hence, ${{a}^{3}},{{b}^{3}},{{c}^{3}}$ are in $G.P.$ is proved.

Note: There is a shortcut to the above question, if a, b, c are in G.P, there is a formula which is ${{b}^{2}}=ac$. Now for proving that ${{a}^{3}},{{b}^{3}},{{c}^{3}}$ is also in $G.P.,$ we can simply substitute $'a'$ by $'{{a}^{3}}'$, $'b'$ by $'{{b}^{3}}'$, $'c'$ by $'{{c}^{3}}'$. Hence, we get ${{b}^{6}}={{a}^{3}}{{c}^{3}}$ which satisfy the fact that ${{a}^{3}},{{b}^{3}},{{c}^{3}}$ is in $G.P.$