# Questions tagged [geometric-series]

For questions about or involving geometric series, a series where successive terms have a common ratio.

379 questions
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### Conditional probability greater than 1. Why?

Let's say I roll a fair die independently many times. Let $X_i$ be the outcome of the $i$th roll. Assume that on the $k$th roll, I get a $1$ or $X_k = 1$. What is the probability that I never rolled ...
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### Why do these expression tend to two? [on hold]

Sorry if I express myself wrongly... Why in a formula like this $x^n-x^{n-1}-x^{n-2}-x^{n-3}-x^{n-4}-...-x^{0}=0$ with $n \to \infty$ $x \to 2$ ?
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### Why are these 2 summations not equal to each other?

Why is:$$\sum_{n=1}^\infty \left(\frac{(-1)^{n-1}(x-1)^n}{n}\right) \neq \sum_{n=0}^\infty \left(\frac{(-1)^{n}(x-1)^{n+1}}{n+1}\right)$$ This question arose when I tried to get the series ...
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### Finding values for range of terms in a geometric sequence

I am NOT math person. I'm a musician who needs some math help. Given the following: A1 = An / (r)^(n-1) I know "An" and "r" and "n"; I want to find "A1" but only for a specific range of terms in ...
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### Solve recurring sequence using a generating function

I have the sequence $a_n=3a_{n-1}-3a_{n-2}+a_{n-3}$, $\forall\ n \ge 3$, with $a_0=2$, $a_1=2$, $a_2=4$ being the known terms, and I want to find a non-recursive equation for $a_n$ using a generating ...
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### How can I simplify this expressions to get one formula?

the expressions are : f(1) = (a^0) f(2) = (a+1) f(3) = (a^2+a+1) and the answer is f(n)= (a^n-1) /a-1, it is the formula for the sum of the geometric series right ? I have tried to find the formula ...
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### Prove that $\sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty$, where $|r_1|, |r_2| < 1$

I'm trying to prove that: $$\sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty$$ where $|r_1|, |r_2| < 1$ and $c_1, c_2$ are some arbitraty real numbers ($r_1, r_2$ are also real numbers). If ...
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Q. A geometric series has first term $4$ and common ratio $r$. Where $0 < r < 1$ The first, second and fourth terms of this geometric series form three successive terms of an arithmetic ...
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### How can I solve this geometrical sequence problem?

How can I solve this problem and can you explain it? $S_n$ is the sum of the first $n$ terms of a geometric sequence with first term $a$ and a common ratio $r$. Let $P_n$ represent the product of ...
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### Sum - geometric series [closed]

I have series a + 2(1-a)a + 3 (1-a) (1-a)a + 4 (1-a)(1-a)a + … Does this series have a common ratio? How can I calculate sum and average?
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### How to calculate the summation of $n \cdot 2^n$? [duplicate]

So I know that you can take the derivative of this and multiply by x and do integrals or something like that. However, I am just wondering if there is a way to come to the summation of the series ...
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I'm asked to find a geometric series for $f(x) = \frac{1}{(1 + x)^2}$, I integrate it first and I get $F(x) = \int{\frac{1}{(1 + x)^2}dx} = \frac{1}{-2(1 - (-x))}$ Since $\frac1{1-x} = \sum_{n=0}^\... 1answer 29 views ### Prove that$\>$for$x\neq1$and k${\in}\> \textbf{Z}_{\geq0}$,$\>\sum_{j=0}^{k}x^{j}=\frac{1-x^{k+1}}{1-x}$. My response was as follows: We may test our base case, that is when$k=0$. This computes to$x^{0}=\frac{1-x^{1}}{1-x}$which equates to$1=1$. Whence$k=0$holds. Therefore, such a statement$P_{k} ...
I have a matrix $A$ such that the spectral radius is $< 1$. It is well known that $I+A+A^2$... converges. Does it then follow that the geometric series of each entry also converges? The matrix ...
A line crosses the $x$ and $y$ axes at $(a,0)$ and $(0,1)$ respectively, where $a>0$. Square are placed successively inside the right angled triangle thus formed. What is the area enclosed by all ...