Prove that (4^𝑛 − 1)/3 is always a positive integer for 𝑛 ∈ ℕ.

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Prove that (4^𝑛 − 1)/3 is always a positive integer for 𝑛 ∈ ℕ.

for this question so far i proved n =1 is valid

so 4-1/3 = 1 which is positive

then assume n = k is true

try for n=k+1

then i have (4^k+1 - 1)/3

this is where i stopped

YEEEEEET Dec 2, 2018

edited by YEEEEEET Dec 2, 2018

CPhill · Answer 1 · 2018-12-02T10:16:34

Here you go, YEEEEEET....

(4^(k + 1) - 1 ) / 3 =

[ 4^(k + 1) - (4 - 3) ] / 3 =

[ 4* 4^k - 4 + 3 ] / 3

[ 4 * 4^k - 4 ]/ 3 + 3/3

4 [ 4^k - 1 ] / 3 + 1

4 [ (4^k - 1) / 3 ] - 1

And since we assumed that (4^k -1) /3 is a positive integer.....then 4 times this is also a positive integer

And adding 1 to this still results in a positive integer....