Problems need HardWork #11

Algebra Level 3

Find the sum of all real number x x for which :

1 0 x + 1 1 x + 1 2 x = 1 3 x + 1 4 x 10^x + 11^x + 12^x = 13^x + 14^x


The answer is 2.

Dev Sharma by Dev Sharma , 20, India

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1 solution

Let 12 12 is a a , and by guessing, pretend x = 2 x=2 because it is prime: ( a − 2 ) 2 + ( a − 1 ) 2 + a 2 = ( a + 1 ) 2 + ( a + 2 ) 2 (a-2)^2+(a-1)^2+a^2=(a+1)^2+(a+2)^2 ⇒ a 2 = 12 a \Rightarrow a^2=12a a = 12 a=12 That's amazing! x ≠ 0 x\not=0 because 3 ≠ 2 3 \not= 2

Well, I can't substitute actually. I just know any odd numbers of consecutive integers with powers of 2.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Prime numbers got nothing to do with it.

Pi Han Goh - 5 years, 9 months ago

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Oh yeah! I forget to mentioned about that.

Adam Phúc Nguyễn - 5 years, 9 months ago

Could there be a proof that doesn't use guessing?

First Last - 5 years, 6 months ago

Yup and actually the same thing can be generalized. a^2+(a+1)^2+(a+3)^2+...(a+k)^2=(a+k+1)^2+(a+k+2)^2+...(a+2k)^2 Then a=k(2k+1) 0ver here this does hold true here so x=2 is the answer

Aditya Kumar - 5 years ago

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