Prove this one If a b 0, n∈N then prove that an bn - Maths - Algebra of Matrices

Question

Prove this one

If a > b > 0, n∈N then prove
that an – bn > n a
- b a b 1 2 n - 1

Zuhaib Ul Zamann · Accepted Answer

Dear student,
The question needs correctionIt shoud be an-bn≥na-babn-12For n=1LHS =a-bRHS=a-bSo it is true for n=1So let it be true for n=k⇒ak-bk≥ka-babk-12Now ak+1-bk+1≥ak-bka+b  {Because bak-abk>0 as a>b}So ak+1-bk+1≥ka-babk-12a+b  Now from AM>GM we have a+b>2abSo ak+1-bk+1≥ka-babk-122ab12ak+1-bk+1≥2ka-bab(k+1)-12As k∈N 2k≥k+1So we have ak+1-bk+1≥k+1a-bab(k+1)-12Inequality is true for n=k+1 and hence by induction is true for alln∈NRegards

Prove this one If a > b > 0, n∈N then prove that an – bn > n a - b a b 1 2 n - 1 .

Prove this one

If a > b > 0, n∈N then prove that aⁿ – bⁿ > $n (a - b) {(a b)}^{\frac{1}{2} (n - 1)}$ .