IIT JEE , AIEEE Forum - Mathematics , Algebra - Give wuth justificationa natural no. n for which 3^9+3^12+3^15+3^n is a perfect cube

mroobm

Give wuth justificationa natural no. n for which 3^9+3^12+3^15+3^n is a perfect cube

sriram.a

yaar very simple take 3^9(1+3^3+3^6+3^{n-9))=x^3
3^9(1+27+729+3^{n-9})=x^3
3^9is a cube 1+27+729+3^{n-9}) should be a perfect cube
757+3^{n-9})=x^3(x=10,11,12,13,14,................)
for x=10 n=14 the entire equatin becomes a perfect cube
for any other x there is no n such thatl equation is a perfect cube

animal

hey dude it is easy

3⁹+3¹²+3¹⁵+3ⁿ=a³ where a $\epsilon$ $\Re$

taking log both sides

9log3+12log3+15log3+nlog3=3loga

3log3+4log3+5log3+n/3log3=loga

log(3³ x 3⁴ x 3⁵x 3^n/3)=loga

taking antilog

3^12+n/3=a

therefore for a to be a perfect cube n should be a multiple of 9.

smallest n=9.

hope u got it.........

sriram.a

hey log cannot be taken in that way.
its log(3^9+3^12+3^15+3^n) is never equal to 9log3+12log3+15log3+nlog3

animal

hey I have not posted that one bcoz i cant do such a blunder some one has posted a rubbish soln. on my name.I need to find him.

sorry for such a rubbish soln. as it is on my name.

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