Asked by Help

Given that 3^9 + 3^12 + 3^15 + 3^n is a perfect cube number. Calculate the
value of n.
Answered by Alyssa
3^36 + ? = perfect cube
? Has to equal 3^?

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Answered by Reiny
3^9 + 3^12 + 3^15 + 3^n
= 3^9(1 + 3^3 + 3^6 + 3^(n-9) )
= 3^9(757 + 3^(n-9))
3^9 is a perfect cube, so all we need is
(757 + 3^(n-9)) to be a perfect cube, clearly n >9

perfect cubes past 757 with 3^(n-9) as the 2nd number:

1000 243 <---- well, that was lucky
1331 574
...

we know that 243 = 3^5
so n-9 = 5
n = 14

check:
if n=14

3^9 + 3^12 + 3^15 + 3^n
= 3^9 + 3^12 + 3^15 + 3^14
= 3^9 (1 + 3^3 + 3^6 + 3^5)
= 3^9 ( (1 + 27 + 729 + 243)
= 3^9 * 1000
= 3^9 * 10^3
= (3^3)^3 * 10^3
= 270^3
which is a perfect cube
Answered by Cgghhji
Fhuthhhj
Answered by lallu bhansali
or be chall haat
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