Question

Prove by mathematical induction 3^n<4^n , n€N

Answer 1

$let \: p(n) = 3 {}^{n } < 4 {}^{n} \\ for \: n = 1 \\ 3 < 4 \\ so \: p(n) \: is \: true \: for \: n = 1 \\ let \: p(n) \: is \: true \: for \: n = k \\ \\ p(k) = 3 {}^{k} < 4 {}^{k } \: \: \: \: \: \: \: eq.1\\ now \: we \: shall \: prove \: that \: p(n) \: is \: true \: for \: n = k + 1 \\ 3 {}^{k + 1 } < 4 {}^{ k+ 1} \\ multiplying \: eq.1 \: by \: 3\: \\ 3 {}^{k} \times 3 < 4 {}^{ k} \times 3 \\ 3 {}^{k + 1} < 4 {}^{k} \times 3 < 4 {}^{k + 1} = 4 {}^{k} \times 4 \\ \\ hence \: \\ 3 {}^{k + 1} < 4 {}^{k + 1} \\ so \: p(n) \: is \: true \: for \: n = k + 1 \\ then \: by \: pmi \: p(n) \: is \: true \: for \: all \: n \: belongs \: to \: n \: \: \: \:$