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Questions tagged [geometric-series]

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

5
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424 views

Expression for sum of $n$ exponentials

So I have this sum of exponentials and I would like to find an expression for it. $$\sum^n_{i=1} e^{\mu(i-1)} $$ Note that $i$ is not an imaginary indicator. I am aware there is a formula for ...
3
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2answers
47 views

Prove that every geometric sequence allows for $S_n(S_{3n}-S_{2n})=(S_{2n}-S_{n})^2$

$S_n(S_{3n}-S_{2n})=(S_{2n}-S_{n})^2$ Is there a way to prove this without expanding everything based on the geometric sum formula? I get lost very easily when trying to solve this conventionally and ...
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2answers
44 views

Convert expression to standard geometric progression

How can I transform a formula describing a curve (a discrete series) to the standard formula for a geometric progression? (Note: Maths is still very much a foreign language to me. I could be using ...
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2answers
26 views

Finding the infinite sum of a geometric series

I have the series: $3 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8}$, I need to calculate the exact value of it. I have found that the equation of this is $a_n = 3(\frac{1}{2})^{n-1}$, but I am not ...
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1answer
22 views

Find the Year of the Question paper it belongs to (Maths 9709 ) [closed]

The first two terms of a geometric progression are where 0<θ<π/2 (i) Find the set of values of θ for which the progression is convergent. [2] Which Year Question paper is this? Year with ...
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0answers
109 views

Show $ \sum_f \lambda(f)t^{\deg f} = \prod_g \big(1 - \lambda(g)t^{\deg g}\big)^{-1} $

So I want to show that $$ \sum_f \lambda(f)t^{\deg f} = \prod_g \big(1 - \lambda(g)t^{\deg g}\big)^{-1} $$ where the sum is over monic polynomials and the product is over all monic irreducible ...
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1answer
25 views

Me possibly overthinking a geometric series question

I am creating a study sheet for geometric series and came across this questions Do you think there is an infinite geometric series with first term 10 and a sum of 4? If so, find one infinite ...
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2answers
29 views

Sum of infinite geometric series with two terms in summation

I have an infinite geometric series: $$\sum_{i=1}^{\infty} \frac{r^i}{(1+d)^i},$$ where $d$ is a constant. I would like to use the sum of infinite geometric series formula, but I cannot see how to use ...
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0answers
30 views

Computing the value of a function based on the geometric series of 1/2^n

So in John B. Conway's First Course in Analysis Book we're told to consider the interval $[a,b],$ and let $\{r_n\}$ denote a sequence of all rationals in said interval. Next, we define a map $\alpha:[...
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1answer
42 views

Complex geometric series $\frac{1}{6i}\sum_{k=0}^{\infty}\left(\frac{z-3i}{6i}\right)^n$

$\dfrac{1}{z+3i}$ can be interpreted as the sum of the geometric series $\displaystyle\dfrac{1}{6i}\sum_{k=0}^{\infty}\left(\frac{z-3i}{6i}\right)^n$ this can be obtained by writing: $\dfrac{1}{z+3i}=\...
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0answers
16 views

Unit step function into geometric series

The unit staircase function is defined as follows: $f(t)=n$ if $n-1 \le t < n$ for $n \in \{1,2,3, \ldots\}$. How do you show that $\displaystyle f(t) = \sum_{k=0}^\infty u(t-k)$?
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2answers
37 views

$\frac{1}{x^2} $ using geometric series vs. Taylor

So I have the function $f(x) = \frac{1}{x^2} $ and I want to represent it as a Taylor series [centered @ x=1] (also evaluating at $x = 1.02$). $$$$I did it using the standard Taylor series method: $...
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0answers
33 views

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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17 views

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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4answers
107 views

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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1answer
52 views

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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2answers
24 views

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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1answer
35 views

Geometric series, TMUA exam question, please help!

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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1answer
38 views

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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2answers
37 views

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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1answer
72 views

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 ...
2
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1answer
24 views

Substitution in Infinite Geometric Series

I'm getting two different answers for the following series... $$\sum_{k=0}^{\infty} q^{2k}$$ With $-1 < q < 1$, I can solve the series by using the infinite geometric sum formula. $$\sum_{k=...
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0answers
19 views

Nth term of geometric series where each term is rounded to two decimal places

I know that the nth term of a geometric series is $$f(n, a_1) = a_n = a_1 r^{n-1}$$ In my case however, since I am dealing with money, I am rounding each term to two decimal places. I calculate the ...
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0answers
41 views

I've a math problem about series but can't find a suitable formula

I'm developing a program that requires a mathematical formula. You can see it below: "1+1.1+1.21+1.331+1.4641+1.6105+1.7715……" The problem is that I want to find the sum of 100 or 1000 numbers in ...
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3answers
32 views

Summation of a geometric sequence from $1$ to infinity for: $(n^{2})\times ((\frac{5}{6})^{n-1})$

I'm fully comfortable with most series and even arithmetico–geometric sequence including n to any exponent if the geometric term is in the form of $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$, and so forth....
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3answers
43 views

Determine the value of $\sum_{k=0}^n i^k$

this is the problem: Determine the value of the sum $\sum_{k=0}^n i^k$ for $n \in \Bbb N$. I started to sum term by term but don't get its value
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0answers
81 views

Can it be said that $\sum_{k=n}^{\infty} q^{k} = \sum_{k=0}^{\infty} q^{n} \cdot q^{k}$?

I would like to know if the following equality is always true for geometric series. $$\sum_{k=n}^{\infty} q^{k} = \sum_{k=0}^{\infty} q^{n} \cdot q^{k}$$ It has worked for me on several occasions, ...
2
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3answers
58 views

Finding a geometric series for $\frac{1}{(1 + x)^2}$

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}^\...
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1answer
27 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} ...
0
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1answer
32 views

Geometric series of entries of matrix with spectral radius smaller

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 ...
3
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1answer
93 views

Series of Squares Under Triangle

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 ...
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1answer
42 views

How would I find the common ratio to determine the sum of the given geometric serie: [closed]

How would I find the common ratio to compute the sum of the given geometric serie: $$\sum_{n=1}^\infty= \frac{(8^n+2^n)}{9^n}$$
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4answers
286 views

How to simplify or upperbound this summation?

I am not a mathematician, so sorry for this trivial question. Is there a way to simplify or to upperbound the following summation: $$ \sum_{i=1}^n{\exp{\left(-\frac{i^2}{\sigma^2}\right)}}.$$ Can I ...
0
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1answer
29 views

A geometric series has second term $6$ and ratio of the seventh term to the sixth term is $3$. What does this question really means?

A geometric series has second term 6 and ratio of the seventh term to the sixth term is $3$. What does this question really means? Sorry I just couldn't get this question. $t(2)=6,$ $6=ar$ from my ...
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0answers
44 views

What is the general series notation for any repeating set of alternating signs?

$$\sum_{n=1}^\infty (-1)^n=-1+1-1+1-1+1\pm\cdots$$ $$\sum_{n=1}^\infty (-1)^{n(n+1)/2}=-1-1+1+1-1-1+1+1\pm\cdots$$ $$\sum_{n=1}^\infty (-1)^{n(n+1)(n+2)(n+3)/8}=-1-1-1-1+1+1+1+1\pm\cdots$$ Generally ...
3
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2answers
72 views

Find the sum of the first n terms if a series is both geometric and arithmetic

I have a simple series of the form: $n+q(n-1)+q^2(n-2)+q^3(n-3)+q^4(n-4)...q^{n-1}$ Where $n$ is in the natural numbers, $q$ is a probability between $[0,1]$. I was wondering if there is some way to ...
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In sum of $n$, $S_n$ =$\frac{a(1-r^n)}{1-r}$, so if $r$ is positive and I wanted to find $n$. How do turn $r$ in to non-negative?

I am given this question find the number of term in this geometric series $3 +1 + \frac{1}{3} , ... , \frac{121}{27}$. So $a=3$ and $r=\frac{1}{3}$ I subbed all the numbers in and I got $\frac{242}{...
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2answers
76 views

Geometric series: two ways to tackle same problem giving me issues

I am trying to get a closed form expression for the following sum: $$x + 2x^2 + 3x^3 + \cdots + nx^n$$ so by perturbation method: $$S = x + 2x^2 + 3x^3 + \cdots + nx^n$$ $$xS= x^2 + 2x^3 + \...
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1answer
20 views

Calculate total length of linear growth equation

Alright, I'm having a really difficult time putting the correct mathematical terms on the problem I'm trying to deal with (which is probably why I can't find an answer), so bear with me.* ...
2
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3answers
47 views

Finding sum of a geometric series

I am asked to find the summation of $1/3^n$ from $n=5$ to infinity. I have done the calculation: $1/(1-r)$, for $r=1/3$, and received $1.5$. As this summation starts from $5$, I subtracted $3^0, 3^-1,...
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2answers
33 views

Calculating the Net Present Value (NPV)

When input the apropriate data is subbed into the equation we get: $$NPV=\sum_{t=0}^\infty\frac{200}{1.1^t}$$ I have been told that the second term looks like $$\sum_{t=0}^\infty\frac{200}{1.1^t} =\...
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1answer
35 views

Concrete Mathematics chapter 2 infinite calculus (geometric series)

Did the authors use finite calculus to evaluate this sum? If so then how? $$ \sum_{k\ge0} x^k = \lim_{n\rightarrow\infty} \frac{1-x^{n+1}}{1-x} = \begin{cases} \frac{1}{1-x} & \text{if $0 \...
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1answer
66 views

Evaluating $\sum_{i_1=1}^{\infty} \sum_{i_2=1}^{i_1-1} \sum_{i_3=1}^{i_2-1} … \sum_{i_m=1}^{i_{m-1}-1} \prod_{k=1}^m a_k^{i_k}$

This is a generalization of my answer to $\sum_{1\leq l \lt m\lt n} \dfrac{1}{5^l3^m2^n}$ I showed there that, if $0 < a, b ,c < 1$, $\begin{array}\\ s(a, b, c) &=\sum_{1\leq l \lt m\lt n} ...
3
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3answers
56 views

Proving this partial sum is bounded

I want to show that the partial sum of $\sum_{n=1}^{\infty} {{e^{inx}}}$ with $x ∈ {(0, 2{\pi})}$ is bounded. I know that the solution is: $|\frac {1-e^{i(n+1)x}}{1-e^{ix}} - 1| \leq \frac {2}{|1-e^{...
9
votes
3answers
423 views

How to find the sum of $1+(1+r)s+(1+r+r^2)s^2+\dots$?

I was asked to find the geometric sum of the following: $$1+(1+r)s+(1+r+r^2)s^2+\dots$$ My first way to solve the problem is to expand the brackets, and sort them out into two different geometric ...
1
vote
1answer
73 views

A KhanAcademy geometric series word problem about doses

Niklas takes a dose of 25 mcg of a certain supplement each day. The supplement has a half life of 4 hours, meaning that 1/64 of the supplement remains in the body after each day. How much of the ...
0
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1answer
26 views

Find the indicated term of the geometric sequence algebraically. [closed]

a1=18, r=4/3, 8th term 1.Find the indicated term of the geometric sequence algebraically.
2
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4answers
37 views

Simplifying a Geometric Series with Two Power Terms

I have derived a geometric series below that I want to simplify but keep making a mess. Can anyone help? $$s = aq^{n-1}r^{0} + aq^{n-2}r^{1} + \dotsb + aq^{1}r^{n-2} + aq^{0}r^{n-1}$$ I get the ...
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2answers
28 views

Convergence of geometric series factor 3/4 word problem

I came across this problem on AoPS: A rubber ball is dropped from a 100 ft tall building. Each time it bounces, it rises to three quarters its previous height. So, after its first bounce it rises to ...
0
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2answers
466 views

How to find the sum of a geometric series with a negative common ratio?

I have a geometric series with the first term 8 and a common ratio of -3. The last term of this sequence is 52488. I need to find the sum till the nth term. While calculating the nth term for 52488 ...