Intereting Posts

Can finite theory have only infinite models?
Nimbers for misère games
Comparison trees
How do we find specific values of sin and cos given the series definition
If matrix A is invertible, is it diagonalizable as well?
Vandermonde identity in a ring
Logarithmic derivative of Riemann Zeta function
The roots of $t^5+1$
Two closest sums of pairs of reciprocals
Nature of the series $\sum\limits_{n}(g_n/p_n)^\alpha$ with $(p_n)$ primes and $(g_n)$ prime gaps
References for Topology with applications in Engineering, Computer Science, Robotics
solution of $y' = \exp \left(-\frac yx\right) + \frac yx$
Colliding Bullets
Show that $x^n + x + 3$ is irreducible for all $n \geq 2.$
“Direct sums of injective modules over Noetherian ring is injective” and its analogue

I need to understand why this :

$$(1+4+\ldots+4^{n−1})\equiv n \pmod3$$

Is that because

- Why not to extend the set of natural numbers to make it closed under division by zero?
- How can I visualize division of negative numbers
- Proving there is no natural number which is both even and odd
- The last digit of $2^{2006}$
- $4494410$ and friends
- “Alternating” sums of fourth powers.

\begin{align}

1&\equiv -2 \pmod3\\

4&\equiv 1 \pmod3\\

4^{2}&\equiv1 \pmod3\\

\ldots&\equiv\ldots\\

4^{n-1}&\equiv1 \pmod3

\end{align}

Am I right? Would you please explain to me more?

- How do you work with the IEEE 754 32-bit floating point format?
- Difference between $\sqrt{x^2}$ and $(\sqrt{x})^2$
- Is there any formula for number of divisors of $a \times b$?
- Can multiplication be defined in terms of divisibility?
- What combinatorial quantity the tetration of two natural numbers represents?
- What is “multiplication by juxtaposition”?
- Why do we use “congruent to” instead of equal to?
- The last digit of $n^5-n$
- Is it possible to simulate a floor() function with elementary arithmetic?
- Problem with my floor…

$$1 \equiv 1 \pmod 3$$

$$4 \equiv 1 \pmod 3$$

$$4^2 \equiv 1 \pmod 3$$

$$\dots$$

$$4^{n-1} \equiv 1 \pmod 3$$

$$1+4+ \dots +4^{n-1} \equiv 1+1+ \dots +1 \equiv n \pmod 3$$

Rewrite $4^n$ as $(3+1)^n$. Then, using the binomial expansion,http://en.wikipedia.org/wiki/Binomial_theorem we get $$3^n+(nC1)3^{n-1} +(nC2)3^{n-2}+…+(nCk)3^{n-k}…(nC(n-1))3 +1^n $$ , where $nCk$ means $\frac {n!}{k!(n-k)!}$ . Notice every term except the last one is divisible by 3 , so that the sum itself –meaning $3^n$ itself , is $1mod3$.

- Question about Geometric-Harmonic Mean.
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- Show that axiom of replacement implies from the Axiom of Specification
- Prove that $\sum\limits_{A\subseteq }\sum\limits_{B\subseteq } |A\cap B|=n4^{n-1}$
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- Number of ordered pairs $(a,b)$ such that $ab \le 3600$
- What is the average of rolling two dice and only taking the value of the higher dice roll?
- Number of subgroups of prime order
- Can we determine uniform continuity from graphs?