Mathematics Stack Exchange
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Questions tagged [geometric-progressions]

A geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence

A geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. This can be useful when evaluating (in)finite series or determining a closed form for a recurrence relation.

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Values of $\sum_{n=0}^\infty x^n$ and $\sum_{n=0}^N x^n$

Why does the following hold: \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*} Can we generalize the above to $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ? Are there some…
AlexBrand
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Why is a geometric progression called so?

Just curious about why geometric progression is called so. Is it related to geometry?
dark32
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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…
akusaja
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Can we calculate $ i\sqrt { i\sqrt { i\sqrt { \cdots } } }$?

It might be obvious that $2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { \cdots } } } } } } $ equals $4.$ So what about $i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { \cdots } } } } } } \text{ ?} $ The answer might be $-1$, but I'm…
user536517
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Double summation with improper integral

So my friend sent me this really interesting problem. It goes: Evaluate the following expression: $$ \sum_{a=2}^\infty \sum_{b=1}^\infty \int_{0}^\infty \frac{x^{b}}{e^{ax} \ b!} \ dx .$$ Here is my approach: First evaluate the integral: $$…
user271938
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Is it possible to cover $\{1,2,...,100\}$ with $20$ geometric progressions?

Recall that a sequence $A=(a_n)_{n\ge 1}$ of real numbers is said to be a geometric progression whenever $\dfrac{a_{n+1}}{a_n}$ is constant for each $n\ge 1$. Then, replacing $20$ with $12$, the following question comes from an old Russian…
Paolo Leonetti
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Arithmetic and geometric sequences: where does their name come from?

Where does the name of these two famous types of sequences come from? The article Geometric progression of Wikipedia says that the geometric sequence is called like this because every term is the geometric mean of its two adjacent terms. Though it…
Taladris
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Cover $\{1,2,...,100\}$ with minimum number of geometric progressions?

In another question, posted here by jordan, we are asked whether it is possible to cover the numbers $\{1,2,\ldots,100\}$ with $20$ geometric sequences of real numbers. Naturally, we would like to extend the question: Problem: What is the minimum…
Colm Bhandal
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Is it more accurate to use the term Geometric Growth or Exponential Growth?

On Wikipedia, the terms Exponential Growth and Geometric Growth are listed as synonymous, and defined as when the growth rate of the value of a mathematical function is proportional to the function's current value but I question whether one term is…
WilliamKF
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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…
O.rka
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Is $1111111111111111111111111111111111111111111111111111111$ ($55$ $1$'s) a composite number?

This is an exercise from a sequence and series book that I am solving. I tried manipulating the number to make it easier to work with: $$111...1 = \frac{1}9(999...) = \frac{1}9(10^{55} - 1)$$ as the number of $1$'s is $55$. The exercise was under…
user69284
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How to find the limit of series? (What should I know?)

There is a couple of limits that I failed to find: $$\lim_{n\to\infty}\frac 1 2 + \frac 1 4 + \frac 1 {8} + \cdots + \frac {1}{2^{n}}$$ and $$\lim_{n\to\infty}1 - \frac 1 3 + \frac 1 9 - \frac 1 {27} + \cdots + \frac {(-1)^{n-1}}{3^{n-1}}$$ There is…
Nikita Hismatov
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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.
oshhh
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Sum of series : $1+11+111+...$

Sum of series $1+11+111+\cdots+11\cdots11$ ($n$ digits) We have: $1=\frac {10-1}9,$ $11=\frac {10^2-1}9$ . . . $11...11= \frac {10^n-1}9$ (number with $n$ digits) and summing them we find the sum ($S$) as: $S=(10^{n+1}-9n-10)/81$ Also the general…
sirous
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Find $a_1+\frac{a_2}{2}+\frac{a_3}{2^2}+\cdots\infty$

A sequence $\left\{a_n\right\}$ is defined as $a_n=a_{n-1}+2a_{n-2}-a_{n-3}$ and $a_1=a_2=\frac{a_3}{3}=1$ Find the value of $$a_1+\frac{a_2}{2}+\frac{a_3}{2^2}+\cdots\infty$$ I actually tried this using difference equation method.Let the solution…
Umesh shankar
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