copy and paste this google map to your website or blog!
Press copy button and paste into your blog or website.
(Please switch to 'HTML' mode when posting into your blog. Examples: WordPress Example, Blogger Example)
Input Form:Deal with this potential dealer,buyer,seller,supplier,manufacturer,exporter,importer
(Any information to deal,buy, sell, quote for products or service)
summation - Prove that $1^3 + 2^3 + . . . + n^3 = (1+ 2 + . . . + n)^2 . . . HINT: You want that last expression to turn out to be $\big (1+2+\ldots+k+ (k+1)\big)^2$, so you want $ (k+1)^3$ to be equal to the difference $$\big (1+2+\ldots+k+ (k+1)\big)^2- (1+2+\ldots+k)^2\; $$ That’s a difference of two squares, so you can factor it as $$ (k+1)\Big (2 (1+2+\ldots+k)+ (k+1)\Big)\; \tag {1}$$ To show that $ (1)$ is just a fancy way of writing $ (k+1)^3$, you need to
Show that $n^3-n$ is divisible by $6$ using induction This answer is with basic induction method when n=1, $\ 1^3-1 = 0 = 6 0$ is divided by 6 so when n=1,the answer is correct we assume that when n=p , the answer is correct so we take, $\ p^3-p $ is divided by 6 then, when n= (p+1), $$\ (p+1)^3- (p+1) = (P^3+3p^2+3p+1)- (p+1)$$ $$\ =p^3-p+3p^2+3p+1-1 $$ $$\ = (p^3-p)+3p^2+3p $$ $$\ = (p^3-p)+3p (p+1) $$ as we assumed $\ (p^3-p) $ is
Proving $1^3+ 2^3 + \cdots + n^3 = \left (\frac {n (n+1)} {2}\right)^2 . . . Hint $ $ First trivially inductively prove the Fundamental Theorem of Difference Calculus $$\rm\ F (n) = \sum_ {k\, =\, 1}^n f (k)\, \iff\, F (n) - F (n\!-\!1)\, =\, f (n),\ \ \, F (0) = 0\qquad$$ The result now follows immediately by $\rm\ F (n) = (n\: (n\!+\!1) 2)^2\ \Rightarrow\ \color {#c00} {F (n)-F (n\!-\!1) = n^3}$ The theorem reduces the proof to a trivial mechanical verification of a
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 is true for the integer 1 but I dont know where to go from there