All Questions
Tagged with geometric-progressions induction
19 questions
-1
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1
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88
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Find sum of $ar^0 + ar^1 + ar^2 + \dotsm + ar^n$
I am trying to deduce the formula of sum of $n$ terms of a GP in a way not described in the book and hence after taking $a$ as common factor, we are left behind with $r^0 + r + r^2 + r^3$ ( I took $n =...
- 361
-1
votes
2
answers
58
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Direct proof using telescoping series for $a_n = 1 + 2^{2-n}$
I am practicing for my math exam, I encountered the following problem:
$a_1=3, 2a_n=1+a_{n-1}$ for $n \geq 2$. Give a proof using mathematical induction and direct proof involving telescoping ...
- 896
1
vote
3
answers
118
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Trouble proving $3^2 + 3^3 + ... 3^n = 9 \cdot \frac{3^{n-1} - 1}2$ by induction
So i'm supposed to prove by mathematical induction that this formula: $3^2 + 3^3 + ... 3^n = 9 \cdot \dfrac{3^{n-1} - 1}2$ holds true for all numbers greater than 2.
I started with the base case and ...
- 13
0
votes
1
answer
285
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Proof by induction summation
I'm a little bit lost. I'm learning Math Discrete, and right now I'm in Proof by Induction. I have a question that uses a summation, and I never saw this kind of question before. Can someone help me? :...
- 15
2
votes
3
answers
85
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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 ...
0
votes
3
answers
99
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Induction with a recursive sequence
Let $(a_{n})_{n \in \mathbb N_{0}}$ be a sequence in $\mathbb Z$, defined as follows:
$a_{0}:=0,
a_{1}:=2,
a_{n+1}:= 4(a_{n}-a_{n-1}) \forall n \in \mathbb N$.
Required to prove: $a_{n}=n2^{n} \...
- 1,838
-2
votes
3
answers
134
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Induction proof of $1+3+\cdots+3^n=\frac{3^{n+1}-1}{2}$ [closed]
How would I prove the following by induction?$$1+3+3^2+3^3+\cdots+3^n=\frac{3^{n+1}-1}{2}$$
for all $n\geq 0.$
I kept trying to create a base case but I am not sure how many I need. I also seem to be ...
- 2,042
1
vote
2
answers
277
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Mathematical Induction with Exponents: $1 + \frac12 + \frac14 + \dots + \frac1{2^{n}} = 2 - \frac1{2^{n}}$
Prove $1 + \frac{1}{2} + \frac{1}{4} + ... + \frac{1}{2^{n}} = 2 - \frac{1}{2^{n}}$ for all positive integers $n$.
My approach was to add $\frac{1}{2^{n + 1}}$ to both sides for the induction step. ...
- 407
4
votes
6
answers
9k
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Prove by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$
My question is:
Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever $n$ is a nonnegative integer.
I'm stuck at the basis step.
If I ...
- 59
3
votes
3
answers
60
views
Induction to prove that for any $r \in \mathbb{R}$ such tht $r \notin (0,1)$ $\sum_{i=1}^n r^i-1 = \frac{(1-r^n)}{1-r}$ for all $n \in \mathbb{N}$.
Use induction to prove that for any $r \in \mathbb{R}$ such that $r \notin (0,1)$ $$\sum_{i=1}^n r^{i-1} = \frac{1-r^n}{1-r}$$ for all $n \in \mathbb{N}$.
My method:
Assume $$\sum_{i=1}^k r^{i-1} = \...
- 307
2
votes
2
answers
160
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Induction proof I'm having trouble with: $1+x+x^2+x^3+...+x^n = \frac{1-x^{n+1}}{1-x}$
So I'm being asked to use induction to prove that for every $x\in\{a\ |\ a\in R, a\neq 1\}$ and for every $n\in N$
$$
1+x+x^2+x^3+...+x^n = \frac{1-x^{n+1}}{1-x}
$$
I have no trouble proving it for $...
3
votes
2
answers
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Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction. [duplicate]
Firstly, this is a homework problem so please do not just give an answer away. Hints and suggestions are really all I'm looking for.
I must prove the following using mathematical induction:
For ...
- 3,027
2
votes
1
answer
4k
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The identity $a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i})$ [duplicate]
How do I use finite induction to prove that
$$a^n-b^n=(a-b) (\sum_{i=0}^{n-1}a^ib^{n-1-i}), \forall a,b\in \Bbb{R}\space \text{and} \space \forall n \in \Bbb{N}?$$
Ok, for $n=2$ it's fine. $a^2-b^2=(a-...
- 2,847
-2
votes
1
answer
96
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Proof by induction of the formula for $2^0+2^1+2^2+...+2^n$? [duplicate]
$2^0+2^1+2^2+...+2^n$ for $n ∈ \mathbb{N} \cup \{0\}$.
I made a conjecture that this is $2^{n+1} - 1$. Now I have to prove it by induction.
I tested the base case where it's equal to zero, and it ...
- 197
-1
votes
1
answer
1k
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Prove by Induction (Geometric Progression): $\sum_{i=0}^n q^i=\frac{q^{n+1}-1}{q-1}$ [duplicate]
Prove by induction that for any real number $q≠1$ and any $n\in \mathbb N$ we have $
\sum_{i=0}^n q^i=\frac{q^{n+1}-1}{q-1}
$
- 3
2
votes
2
answers
111
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Help with Elementary number theory please
Use the second principle of finite induction to establish that for all $n\geq1$ :
$$a^n-1=(a-1)\left(a^{n-1}+a^{n-2}+a^{n-3}+\cdots+a+1\right) $$
Step by step explanation please! I'm confused how the ...
- 2,615
4
votes
2
answers
4k
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Proving the geometric sum formula by induction
$$\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}$$
I wanted to factor a $q^{...
- 503
5
votes
4
answers
349
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Prove that $1+a+a^2+\cdots+a^n=(1-a^{n+1})/(1-a)$.
I have problem. Prove this using Mathematical Induction. I am a newbie in Mathematics. Please help me.
$$1+a+a^2+\cdots+a^n = \frac{1-a^{n+1}}{1-a}$$
This is my way for get the proof
Basic ...
user108200
16
votes
4
answers
17k
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Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
This question is from [Number Theory George E. Andrews 1-1 #3].
Prove that $$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}+...
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