Linear-O-Exponential Equation!

Number Theory Level 5

7 n − 12 2 n + 2 n − 14 3 n + 24 n 6 n = 1 \large{\dfrac{7n-12}{2^n} + \dfrac{2n -14}{3^n} + \dfrac{24n}{6^n} = 1}

Find the sum of all integers n n , such that the above equation satisfies.


The answer is 4.

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Satyajit Mohanty by Satyajit Mohanty , 24, India

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2 solutions

Bolin Chen
Aug 15, 2015

We can know that n≡4(mod 6)and n>0 easily.n can't be too large.So we try n=4 or 10.Then find 4 is satisfied and 10 is too large.

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Siddharth Iyer
Aug 10, 2015

Each of the parts in the addition in the left hand side are less than 1/3 for n>=4 and n<=0, for all n, n>=7 and n<=1 respectively ( basically (7n-12)/(2^n), (2n-14)/(3^n), 24/(6^n) are all less than 1/3 for those specified intervals). All 3 are not less than 1/3 for 0<=n<=6 testing each value gives n = 4

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