# 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

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### How to get sum of $\frac{1}{1+x^2}+\frac{1}{(1+x^2)^2}+…+\frac{1}{(1+x^2)^n}$ using mathematical induction

Prehistory: I'm reading book. Because of exercises, reading process is going very slowly. Anyway, I want honestly complete all exercises. Theme in the book is mathematical induction. There were ...
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### Relation between two series

Consider the two series , A=Σ(2ⁿ/n!) from 1 to ∞. and, B=Σ(4ⁿ/n!) from 1 to ∞. What is the relationship between them?( If any) I think the exponential series might come in handy but the numerator ...
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### How to relate two series(GP and AP) using a positive real number, who have nothing in common?

Here is a question from my book: Given a GP and an AP with positive terms $a,a_1,a_2,a_3...a_n$ and $b,b_1,b_2,b_3,...b_n$ respectively. The common ratio of the GP is different from $1$. Then ...
4answers
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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 ...
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### Annuities with payments in geometric progression

I am having trouble understanding how to solve problems with varying annuities. There is this problem I was given as a homework which I can't figure out. Barry presently has 2.9 million dollars in ...
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### What is the closed form of $\sum_{j=i}^{n} {j}$?

How can I get a closed form from a summation like this? $$\sum_{j=i}^{n} {j}$$ I don’t know how to proceed since the base of the summation is a variable.
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### Solving summations by upper bound - lower bound + 1

I’ve seen many different summations that are solved by the upper bound minus the lower bound plus one. For example: $$\sum_{i=1}^{n} {1} = n-1+1$$ May I always resolve summations in this way or ...
3answers
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### How to compute the time complexity of a triple nested loop represented by $\sum_{i=1}^{2n} \sum_{j=1}^{n} \sum_{k=j}^{n} i-j$

var r = 0; for(var i=1; i<=2*n; i++) { for(var j=1; j<=n; j++) { for(var k=j; k<=n; k++) { r = r + (i - j); } } } I'm trying to use ...
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### Find minimum of $[ x^{(\ln y-\ln z)} + y^{(\ln z-\ln x )} + z^{(\ln x-\ln y)} ]$

If $$x\gt0,y\gt 0, z\gt 0$$ then find the minimum value of $$\left[ x^{(\ln y-\ln z)} + y^{(\ln z-\ln x )} + z^{(\ln x-\ln y)} \right]$$ Today I came across this question ( maybe it's from ...
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### Calculate the sum of geometrical progression

I have the following progression $$\frac{1}{1+x^2} + \frac{1}{(1+x^2)^2} + ... + \frac{1}{(1+x^2)^n}$$ I have that $a=\frac{1}{1+x^2}$ and $q=\frac{1}{1+x^2}$, then using $a\frac{1-q^{n+1}}{1-q}$...
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### Sequences and geometric progressions

Any sequence of natural numbers that contains an infinite arithmetic progression (AP) must have a positive lower density and this alone rules out many candidates (squares, primes, etc.). On the other ...
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### What is the general formula for expansion of 1 + x^(odd number) in terms of (1+x)

I believe there's formula to write $1+x^{2n+1}$ in terms of $(1+x)(\cdots)$ just like how $1+x^3$ can be written as $(1+x)(x^2 -x +1)$. I am not able to find explanation of it anywhere over the ...
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### How to explain mortgage monthly payment formula using school math?

$P = L*\frac{x*(1+x)^n}{(1+x)^n - 1}$ where P - monthly payment L - loan amount x - monthly interest rate n - number of payments Here is in ...
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### $a, b, c$ form a geometric sequence and $\log_c a, \log_ b c, \log_a b$ form an arithmetic sequence.

The common difference of the arithmetic sequence can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ So far, I rearranged the sequences to be 1: $a$,...
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### What is $\sum_{i = 0}^{n} (2^{ki})$? [closed]

What is the result of $\sum\limits_{i = 0}^{n}(2^{ki})$ Is it $\frac{1-2^{kn}}{1-2^k}$?
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### rank of a rectangular Vandermonde matrix

Let the $m\times (n+1)$ rectangular Vandermonde matrix be $V$. More specifically, the matrix $V$ has the following form. \$V=\begin{pmatrix} 1 & a_1 & \cdots & a_1^{n} \\ 1 & a_2 &...