Let (n_k)_{k=1}^{infty} be a sequence of natural numbers such that n_k < n_{k+1} for all k in mathbb{N}. Using induction show that n_k ge k for all k in mathbb{N}.
Added by Ronald H.
Close
Step 1
Since the sequence is of natural numbers, \(n_1\) is a natural number and is greater than or equal to 1. Therefore, the base case is true. ** Show more…
Show all steps
Your feedback will help us improve your experience
Ben Blakesley and 98 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Use mathematical induction to show that $$x^{n}-1=(x-1)\left(1+x+x^{2}+\cdots+x^{n-1}\right)$$for all natural numbers $n$
Additional Topics in Algebra
Mathematical Induction
Use mathematical induction to show that the given statement is true. If $x > -1$. then $(1+x)^{n} \geq 1+n x$ for all natural numbers $n$.
Sequences and Series
Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$. $$4+3+2+\dots+(5-n)=\frac{1}{2} n(9-n)$$
Sequences; Induction; the Binomial Theorem
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD