Question

The number ${{3}^{9}}+{{3}^{12}}+{{3}^{15}}+{{3}^{n}}$ is a perfect cube of an integer for natural number n, then n is
A.12
B.13
C.14
D.15

Answer 1

Hint: Start by taking ${{3}^{9}}$ common from all the terms given in the question and try to compare rest of the terms with ${{\left( 1+x \right)}^{3}}=1+{{x}^{3}}+3{{x}^{2}}+3x$ . As ${{3}^{9}}$ is a perfect cube, so it is necessary that the left part is also a perfect square so it must boil down to the form of ${{\left( 1+x \right)}^{3}}$ .

Complete step-by-step answer:
Let us start the solution to the above question by taking ${{3}^{9}}$ common from all the terms of the given expression.
${{3}^{9}}+{{3}^{12}}+{{3}^{15}}+{{3}^{n}}$
$={{3}^{9}}\left( 1+{{3}^{3}}+{{3}^{6}}+{{3}^{n-9}} \right)$
Now we know that ${{3}^{9}}$ is a perfect cube. So, for the whole expression to be a perfect cube, $\left( 1+{{3}^{3}}+{{3}^{6}}+{{3}^{n-9}} \right)$ must be a perfect cube as well.
Also, if we compare $\left( 1+{{3}^{3}}+{{3}^{6}}+{{3}^{n-9}} \right)$ with the expansion of ${{\left( 1+x \right)}^{3}}=1+3x+{{x}^{3}}+3{{x}^{2}}$ , we find that the first three terms are satisfied if $x={{3}^{2}}$ . Also, this will make the term to be a perfect cube. So, if we put $x={{3}^{2}}$ in ${{\left( 1+x \right)}^{3}}=1+3x+{{x}^{3}}+3{{x}^{2}}$ , we get
${{\left( 1+{{3}^{2}} \right)}^{3}}=1+3\times {{3}^{2}}+{{\left( {{3}^{2}} \right)}^{3}}+3\times {{\left( {{3}^{2}} \right)}^{2}}=1+{{3}^{3}}+{{3}^{6}}+{{3}^{5}}$
 Therefore, we can see that ${{3}^{5}}={{3}^{n-9}}$ , which implies 5=n-9. So, solving this equation in n, we get
$n-9=5$
$\Rightarrow n=14$
Hence, the answer to the above question is option (c).

Note:The above question is not very loud, it is mostly based on your analysing and imagination skills. However, one thing you can notice in such questions is the number of terms and the parts common in each term, as 3 was common in each term other than 1 which was pointing towards the expansion ${{\left( 1+x \right)}^{3}}=1+3x+{{x}^{3}}+3{{x}^{2}}$ . Also, be very careful about the formulas that you are using.