summation - Prove that $1^3 + 2^3 + ... + n^3 = (1+ 2 + ... + n)^2 ...
Thus by the principle of mathematical induction 13 +23 +33 + ⋯ +n3 = (1 + 2 + 3 + ⋯ + n)2 1 3 + 2 3 + 3 3 + + n 3 = (1 + 2 + 3 + + n) 2 for each n ∈N n ∈ N.
induction - Prove that $ n^3 + 5n$ is divisible by 6 for all $n\in ...
So n=3k+1, the remainder of n3 n 3 when divided by 3 is 1 and for 5n 5 n it will be 2 2 and 2 + 1 = 3 2 + 1 = 3. And for n = 3k + 2 n = 3 k + 2, the n3 n 3 leaves remainder 2 when divided by 3 and 5n 5 n …
Proving $1^3+ 2^3 + \\cdots + n^3 = \\left(\\frac{n(n+1)}{2}\\right)^2 ...
Dec 9, 2014 · The result now follows immediately by F(n) = (n(n + 1)/2)2 ⇒ F(n) − F(n − 1) = n3 F (n) = (n (n + 1) / 2) 2 ⇒ F (n) F (n 1) = n 3 The theorem reduces the proof to a trivial mechanical …
how to prove that $f(n)=n^3+n\\log^2n$ = $\\theta(n^3)$?
Nov 21, 2018 · 1 i have f(n) = n3 +nlog2n f (n) = n 3 + n log 2 n and i was trieng to prove that f(n) = n3 +nlog2n f (n) = n 3 + n log 2 n = θ(n3) θ (n 3). but i feel that i am doing it all wrong , which means i …
Use mathematical induction to prove that $n^ 3 − n$ is divisible by 3 ...
Use mathematical induction to prove that n3 − n n 3 n is divisible by 3 whenever n is a positive integer. Ask Question Asked 9 years, 7 months ago Modified 7 years, 7 months ago
geometry - Area identities for a triangular "prism" circumscribed about ...
Jan 1, 2026 · Consider a convex triangular "prism" in $\\mathbb{R}^3$ circumscribed about a sphere. The three lateral faces are planes tangent to the sphere whose outward normals lie in the …
Computational complexity of the Cholesky factorization
Feb 11, 2021 · There are various methods for calculating the Cholesky decomposition. The computational complexity of commonly used algorithms is O(n3) O (n 3) in general.The algorithms …
Series convergence test, $\sum_ {n=1}^ {\infty} \frac { (x-2)^n} {n3^n}$
Jun 28, 2020 · By the ratio test, every x value between -1 and 5 would make the series converge. we just need to find out whether x=-1, 5 makes it converge. x=-1: The series will look like this. $$\sum_ …
elementary number theory - Mathematics Stack Exchange
Let n^3+2n = P (n). We know that P (0) is divisible by 3. The inductive step shows that P (n+1) = P (n) + (something divisible by 3). So if P (0) is divisible by 3, then P (1) is divisible by 3, and then...
Prove that $2^n3^ {2n}-1$ is always divisible by 17
7 Prove that $2^n3^ {2n} -1$ is always divisible by $17$. I am very new to proofs and i was considering using proof by induction but I am not sure how to. I know you have to start by verifying the statement …