# Thread: mathematical induction - inequalities

1. ## mathematical induction - inequalities

hi guys.. exams is in a few days need help on this topic!

can i know how do i solve these questions?

Q: Prove that the following inequalities hold for all n ∈ N.

1. $\displaystyle (1+x)^n\geq1+nx$ if $\displaystyle x\geq-1$

2. $\displaystyle 1^3+2^3+...+(n-1)^3 < \frac{1}{4}n^4 < 1^3+2^3+ ... +n^3$

and one more question on mathematical induction..

Q: Prove by induction that the following statement is true for every positive integer n.

$\displaystyle x-y$ is a factor of $\displaystyle x^n - y^n$

1. $\displaystyle (1+x)^n\geq1+nx$ if $\displaystyle x\geq-1$
• Prove that $\displaystyle (1+x)^0\geq 1+0\times x$ (easy )
• Assume that there exists en integer $\displaystyle n$ such that $\displaystyle (1+x)^n\geq1+nx$ (1). As we want to show that $\displaystyle (1+x)^{n+1}\geq1+(n+1)x$, you may try using $\displaystyle (1+x)^{n+1}=(1+x)(1+x)^n$. (expand this and apply (1))
2. $\displaystyle 1^3+2^3+...+(n-1)^3 < \frac{1}{4}n^4 < 1^3+2^3+ ... +n^3$
Notice that $\displaystyle \frac{n^4}{4}=\int_0^nt^3\,\mathrm{d}t$. As $\displaystyle t\mapsto t^3$ is an increasing function, $\displaystyle \int_{k-1}^{k}(k-1)^3\,\mathrm{d}t<\int_{k-1}^{k}t^3\,\mathrm{d}t< \int_{k-1}^{k}k^3\,\mathrm{d}t$ which can be rewritten $\displaystyle (k-1)^3<\int_{k-1}^{k}t^3\,\mathrm{d}t< k^3$. Can you use this to show the two inequalities ?