Intereting Posts

Negative binomial distribution – sum of two random variables
Generalized Laplace–Beltrami operators
Can you determine a formula for this problem?
Test for the convergence of the sequence $S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$
proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
convergence of a series involving $x^\sqrt{n}$
Evaluating real integral using residue calculus: why different results?
Intuitive Aproach to Dolbeault Cohomology
Are there two different ways to generalize the orthogonal group?
A limit question (JEE $2014$)
On the set of the sub-sums of a given series
Possible mistake in Folland real analysis?
Does convergence in H1 imply pointwise convergence?
Proof of the relation $\int^1_0 \frac{\log^n x}{1-x}dx=(-1)^n~ n!~ \zeta(n+1)$
How do I differentiate this integral?

$$\sum_{r=0}^{k-1}4^r$$

Hi, I was wondering whether anyone could explain how to work this out. I know the end result is $\frac{4^k-1}{3}$, but I don’t know why or how to get there.

Thank you ðŸ˜€

- Computing the sum $\sum_{n=2}^{2011}\sqrt{1+\frac{1}{n^2}+\frac{1}{(n-1)^2}}$
- Limit of alternating sum with binomial coefficient
- If $a+b+c=3$, find the greatest value of $a^2b^3c^2$.
- How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?
- How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?
- Criteria for swapping integration and summation order
- Which series converges the most slowly?
- Evaluation of Indefinite Integral resulting in Hypergeometric Function
- The equivalence of an integral and a sum of integrals
- How to show that $1 \over \sqrt{1 - 4x} $ generate $\sum_{n=0}^\infty \binom{2n}{n}x^n $

Multiply it by $(4-1)$. Expand without turning it into $4-1=3$. a lot of powers of $4$ will simplify except $4^k$ and $1$. Afterwards divide by $4-1=3$.

- How to find the boundary curve of a surface, like the Möbius strip?
- “Intrinsic” treatment of projective spaces
- Does local convexity imply global convexity?
- Limit of $L_p$ norm as $ p \rightarrow 0$
- The prime number theorem and the nth prime
- What is a primary decomposition of the ideal $I = \langle xy, x – yz \rangle$?
- Prove inequality$\sum\limits_{n|i+j+k}x_{i}y_{j}z_{k}\le n^2$
- Euler-Mascheroni constant expression, further simplification
- Let $\alpha,\beta$ be the distinct positive roots of the equation $\tan x=2x$,then find $\int_{0}^{1}\sin \alpha x \sin \beta x$dx
- Showing that $n$ exponential functions are linearly independent.
- For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
- Derangements with repetitive numbers
- Evaluation of $\lim_{n\rightarrow \infty}\sum_{k=1}^n\sin \left(\frac{n}{n^2+k^2}\right)$
- Why can 2 uncorrelated random variables be dependent?
- Definition of the gradient for non-Cartesian coordinates