Tagged Questions
0
votes
1answer
69 views
Why is this summation formula wrong?
This is the alternate form of the summation formula:
$$
\sum^{n}_{k=0} a(c)^k = \frac{ac^{n+1} - a}{c - 1}
$$
so why is this wrong?
$$
\sum^{n}_{k=0} (-\frac{1}{2})^k = \frac{(-\frac{1}{2})^{n+1} - ...
3
votes
6answers
67 views
Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?
How can I show that
$$
\sum_{i=1}^{n-1}i=\binom{n}{2}?
$$
This is what I have tried, but I do not know if it is correct:
Proof.
Let $n=2$. Then,
$$
\begin{align}
\sum_{i=1}^{1}i&=1\text{, ...
2
votes
3answers
59 views
problem with induction?
I am a bit new to logical induction, so I apologize if this question is a bit basic.
I tried proving this by induction:
$$\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$$
Starting with the base ...
1
vote
3answers
73 views
Using induction to verify a statement
I have to prove that this statement is true.
For $n = 1, 2, 3, ...,$ we have $ 1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)/6$
Basically I thought I'd use induction to prove this. Setting $n = p+1$, I ...
9
votes
2answers
127 views
Proving that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$
How do I prove that:
$$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$$
Do I use induction?
0
votes
2answers
70 views
I need help with proofs using mathematical Induction
I need help with this problem: $2+7+12+17+...+(5n-3)=(\frac{n}{2})(5n-1)$
6
votes
4answers
145 views
Mathematics Induction
Question:
Prove by mathematical induction that $$(1)+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)=\frac{1}{6}n(n+1)(n+2)$$ is true for all positive integers n.
Attempt:
I did the the induction steps and I ...
3
votes
3answers
94 views
Induction proof of $\sum_{j=0}^n{(-1)^j{n \choose j}\prod_{k=m+1}^{m+n-1}{(j+k)}}=0$
Does anynone have some hints to prove the following equation by induction for all $n\geq 1$ and $m\in\mathbb{Z} $
$$\sum_{j=0}^n{(-1)^j{n \choose j}\prod_{k=m+1}^{m+n-1}{(j+k)}}=0$$
use for ...
4
votes
2answers
71 views
$\sum_{i=1}^n i\cdot i! = (n+1)!-1$ By Induction
I am trying to prove the following by Mathematical Induction:
$$\sum_{i=1}^n i\cdot i! = (n+1)!-1\quad\text{for all integers $n\ge 1$}$$
My proof by Induction follows:
First prove $P(1)$ is true,
...
4
votes
2answers
292 views
Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$
I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$
I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively ...
3
votes
4answers
59 views
How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$? [duplicate]
How can I prove that $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$? I noticed that in the sum, the denominator has terms that cancel out, but I'm not sure how to take advantage of that.
5
votes
3answers
538 views
Complete induction
I am very confused with complete induction. Because in every task there is something different to do, and I never know what to insert (thats my biggest problem).
Here's the example:
Proof with ...
2
votes
5answers
126 views
Proof via Induction for A Summation
I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically:
$$
...
2
votes
2answers
69 views
Summing $r(r+3)$ using induction
We want to prove the following summation by induction:
$$\sum_{r=1}^{n}r(r+3)=\frac{1}{3}n(n+1)(n+5)$$
The problem is posted for a friend, but others can look at the solution if they want/need.
4
votes
4answers
143 views
Help in proving $ \left( 1 + \frac{1}{n} \right)^{n} \leq \sum\limits_{k=0}^{n} \frac{1}{k!} < 3 $.
I am trying to prove this statement for all $ n \geq 1 $ using induction:
$$
\left( 1 + \frac{1}{n} \right)^{n} \leq \sum_{k=0}^{n} \frac{1}{k!} < 3.
$$
I said:
Base case $ n = 1 $:
$$
\left( ...
1
vote
3answers
96 views
Prove equation with induction
Prove by induction that for every natural number $n\in\mathbb{N}$ and every real number $x\in\mathbb{R}$:
$$(1+x)(1+x^2)(1+x^4)\cdot\,\dots\,\cdot(1+x^{2^{n-1}}) = 1 +x +x^2+\dots+(x^{2^n-1})$$
I ...
6
votes
4answers
112 views
How to “make up a formula” of a sum?
I was asked the following:
Determine a formula for the following sum with $n\in\mathbb N$:
$$S=2+7+12+\cdots+(5n-3)$$
I had no clue about what to do about it. I just wrote, for the sake of doing ...
0
votes
0answers
52 views
Induction Proof of $\sum_{i=1}^{n} (2i)^3 = 2n^2(n+1)^2$ [duplicate]
Possible Duplicate:
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
I want to prove: $\sum_{i=1}^{n} (2i)^3 = 2n^2(n+1)^2$
What I have so far:
Assume ...
1
vote
2answers
49 views
elementary prove thru induction - dumb stumbling
i am trying to prove this statement for all $n \in \mathbb{N}$ with the help of induction:
$4 \sum_{k=1}^{n} (-1)^kk=(-1)^n(2n+1)-1$
base case: n=1
$4 \sum_{k=1}^{1} (-1)^11=-4=(-1)^1(2*1+1)-1$ .. ...
1
vote
1answer
108 views
Expressing $\int \tan^n x\,dx$ with a sum
I was playing around with integrals of $\tan x$, because I knew that both $\int\tan x\,dx$ and $\int\tan^2x\,dx$ were solvable. I then came across the fact that
$$\begin{align}
\int \tan^n x\,dx ...
2
votes
2answers
159 views
Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$
Prove the following statement $S(n)$ for $n\ge1$:
$$\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$$
To prove the basis, I substitute $1$ for $n$ in $S(n)$:
$$\sum_{i=1}^11^3=1=\frac{1^2(2)^2}{4}$$
Great. ...
0
votes
3answers
187 views
How to prove by induction that $\sum^n_{i=1}2^{i-1}=2^n-1$? [duplicate]
Possible Duplicate:
How do I prove this by induction? (sum of powers of 2)
Summation equation for $2^{x-1}$
How can I prove the following by induction?
$$
\sum^n_{i=1}2^{i-1}=2^n-1
$$
I ...
0
votes
2answers
411 views
Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n - 1$ by induction
I'm trying to apply an induction proof to show that $((n+1) 2^n - 1 $ is the sum of $(i 2^{i-1})$ from $0$ to $n$.
the base case: L.H.S = R.H.S
we assume that $(k+1) 2^k - 1 $ is true.
we need to ...
8
votes
4answers
286 views
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
6
votes
6answers
799 views
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing.
$$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$
...
