Articles of geometric progressions

Geometric Progression: How to solve for $n$ in the following equation $\frac {5^n-1}4 \equiv 2 \pmod 7$

Solve $$\frac {5^n-1}4 \equiv 2 \pmod 7$$ How to find the minimum value of $n$ that can satisfy above equation?

Can you prove that $\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}… = \frac{1}{n-1}$?

I’m a student and, while playing with my calculator, find out that: $$\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}…. = \frac{1}{n-1}$$ is it a well defined equivalence and what is its name, is there a proof for that? if we put it this way: $$1+\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}…. = \frac{n}{n-1}$$ what do you call the last term (the sum), the complementary inverse or reciprocal […]

If $a+b+c=3$, find the greatest value of $a^2b^3c^2$.

If $a+b+c=3$, and $a,b,c>0$ find the greatest value of $a^2b^3c^2$. I have no idea as to how I can solve this question. I only require a small hint to start this question. It would be great if someone could help me with this.

How to find a general sum formula for the series: 5+55+555+5555+…?

I have a question about finding the sum formula of n-th terms. Here’s the series: $5+55+555+5555$+…… What is the general formula to find the sum of n-th terms? My attempts: I think I need to separate 5 from this series such that: $5(1+11+111+1111+….)$ Then, I think I need to make the statement in the parentheses […]

Why is $\sum_{n=0}^{\infty }\left ( \frac{1}{2} \right )^{n}= 2$?

This question already has an answer here: Value of $\sum\limits_n x^n$ 5 answers Why $ \sum_{k=0}^{\infty} q^k $ sum is $ \frac{1}{1-q}$ when $|q| < 1$ [duplicate] 3 answers

Why $ \sum_{k=0}^{\infty} q^k $ sum is $ \frac{1}{1-q}$ when $|q| < 1$

This question already has an answer here: Value of $\sum\limits_n x^n$ 5 answers

Value of $\sum\limits_n x^n$

Why does the following hold: \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3 ? \end{equation*} Can we generalize the above to $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ? Are there some values of $x$ for which the above formula is invalid? What about if we take only a finite number of terms? Is there a simpler formula? $\displaystyle […]

Proving the geometric sum formula by induction

$$\sum_{k=0}^nq^k = \frac{1-q^{n+1}}{1-q}$$ I want to prove this by induction. Here’s what I have. $$\frac{1-q^{n+1}}{1-q} + q^{n+1} = \frac{1-q^{n+1}+q^{n+1}(1-q)}{1-q}$$ I wanted to factor a $q^{n+1}$ out of the second expression but that 1- is screwing it up…

Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$

This question is from [Number Theory George E. Andrews 1-1 #3]. Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$ This problem is driving me crazy. $$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1)}$$ $(x^n-y^n)/(x-y) =$ the sum for the first $n$ numbers and then I added $(xy^{(n+1)-2}+y^{(n+1)-1})$ which should equal $(x^{n+1}-y^{n+1})/(x-y)$ but I can’t figure it out This is a similar problem in the […]

Summation equation for $2^{x-1}$

Since everyone freaked out, I made the variables are the same. $$ \sum_{x=1}^{n} 2^{x-1} $$ I’ve been trying to find this for a while. I tried the usually geometric equation (Here) but I couldn’t get it right (if you need me to post my work I will). Here’s the outputs I need: 1, 3, 7, […]