# Tagged Questions

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### Finding a formula for $1+\sum_{j=1}^n(j!)\cdot j$ using induction

I need help with finding the formula and proving it by induction. Am stuck, but the professor says we should know this by now.
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### Proof by Induction [Number Theory by George E. Andrews 1-1 #2] [duplicate]

I am to use mathematical induction to prove that: $$1^3 + 2^3 + 3^3 + \cdots + n^3 = (1 + 2 + 3 + \cdots + n)^2$$
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### Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
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### Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
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### Summation Induction when lower limit is not 1

The question is use induction to prove that $$\sum_{r=2}^n (r^2+r+1)r! = (n+1)^2n!-4$$ I don't understand how to even get the P1 statement since when I substitute r = 2 into the LHS and n = 1 into ...
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### Proof via induction $1\cdot3 + 2\cdot4 + 3\cdot5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}$

(b) Prove that for every integer $n \ge 1$, $$1\cdot3 + 2\cdot4 + 3\cdot5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}$$ This is the second part of a two part question. Part (a) was the following: ...
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### Discrete Math Induction: $\sum^n_{i=1} \frac1{i(i+1)}$ [duplicate]

For $\sum^n_{i=1} \frac1{i(i+1)}$ Find a formula and proofs that it holds for all n ≥ 1. How would I find the formula for this one that can hold for all n ≥ 1?
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### Discrete Math On Induction proof: $\sum_{i=1}^n n2^n = (n-1)2^{n+1} + 2$ [duplicate]

Show by induction that the following formulas hold. $\sum_{i=1}^n n2ⁿ = (n-1)2^{n+1} + 2$ What did a similar problem to this but this one is a little different. I think is because this one has a ...
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### Using induction to prove $\sum\limits^n_{k=1} 9^k = 0.5 \cdot \sum\limits^{2n}_{k=1} (-1)^k \cdot 3^{k+1}$

$$\sum^n_{k=1} 9^k = 0.5 \cdot \left[\sum^{2n}_{k=1} (-1)^k \cdot 3^{k+1}\right]$$ I have tested both with a python script and it seems to be correct. For the life of me, I am unable to unwind the ...
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### Proving a summation involving binomial coefficients.

I need to prove the following inductively: (http://upload.wikimedia.org/math/9/e/5/9e57871ba17c1ad48e01beb7e1bb3bb9.png) $$\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$$ And for the life of me I can't ...
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### How can I come up with a formula for this summation?

I have to come up with a formula for: $$\sum_{0\le i\le n\text{, i is even}}^\ i^2$$ and then prove it by using induction. I know how to do the proof, but I am stuck on coming up with the formula. I ...
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### Mathematical Induction (summation): $\sum^n_{k=1} k2^k =(n-1)(2^{n+1})+2$

I am stuck on this question from the IB Cambridge HL math text book about Mathematical induction. I am sorry about the bad formatting I am new and have no idea how to write the summation sign. Using ...
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### Induction: show that $\sum\limits_{k=1}^n \frac{1}{\sqrt{k}} < 2 \sqrt{n}$ for all n $\in Z_+$

The question: show by using induction that $\sum\limits_{k=1}^n \frac{1}{\sqrt{k}} < 2 \sqrt{n}$ for all n $\in Z_+$ My attempt at a solution: The base case $n = 1$ is true. First we use the ...
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### Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$

My question is: show $$\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$$ $$n\geq m\geq 1$$ I tried to do this via induction and failed. there has to be another way of doing this. We could either ...
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### Use induction and Newton's binomial formula to show that $\binom{n}{0}+\binom{n}{1}+\cdot+\binom{n}{n}=2^n, \forall n\in \mathbb N$ [duplicate]

Use induction and Newton's binomial formula to show that: $i)$ $\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n}=2^n, \forall n\in \mathbb N$ $ii)$ ...
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### Certain step in the induction proof $\sum\limits_{i=0}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$ unclear

It's about proving the following: $$\sum\limits_{i=0}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$ I understand every step in the master solution, however, I have no idea how one can know by intuition to ...
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### Binomial coefficients identity (sum of the powers of the natural numbers)

I've found exercise with binomial coefficients in Kostrikin's book. Proof that $\sum_{i=1}^n{{r+1}\choose{i}}\left(1^i+2^i+\dots+n^i\right)=(n+1)^{r+1}-(n+1)$ I was trying to check that for ...
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### Given n $\in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$

I tried to solve it using induction, but that got me no were, in the middle of the equation stat appearing ks that I don't see how to get out of the equation. I think the easiest way to prove it, it's ...
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### Prove that $\sum_{k=0}^n\frac{1}{k!}\geq \left(1+\frac{1}{n}\right)^n$ [duplicate]

It basically says it all in the title. I tried solving the inequality using the bernoulli inequality somehow $$\dfrac{\displaystyle\sum_{k=0}^n\frac{1}{k!}}{(1+\frac{1}{n})^n}\geq 1,$$ but the ...
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### Prove a summation inequality by induction

I was having trouble proving by induction with this problem. $$\sum_{i=1}^n \frac{3}{4^i} < 1$$ for all $n \geq 2$ I went to see my professor and he said try proving this equality ...
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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 ...
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### How do you solve a recurrence with a summation function inside

Show that $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$ is the same as $$t(n) = 2^n$$ Initial condition $t(0) = 1$
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I have this exercise by my professor that I have no idea how to solve. Any help would be greatly appreciated: Using the method of mathematical induction show that for all $n \geq 1$, $n ... 2answers 56 views ### Prove$\sum_{i=1}^{n}i\left(\begin{array}{c} n\\ i \end{array}\right)=n2^{n-1}$using induction. I have already derived the formula$\sum_{i=1}^{n}i\left(\begin{array}{c}n\\i \end{array}\right)=n2^{n-1}$directly just by doing some algebraic manipulations to the summand, which is indeed proves ... 2answers 85 views ### How do I prove$2^{n+1} + 2n + 1 = 2^{n+2} - 1$I am attempting to prove using induction:$\sum_0^n 2i = 2^{n + 1} - 1$I have gotten to the point where I need to show:$2^{n+1} + 2n + 1 = 2^{n+2} - 1$How do I prove this? Or should I be ... 1answer 66 views ### Proof by induction that$1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$[duplicate] How would I go about solving this question? Use induction to prove that for all integers$n ≥ 1$, $$1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$$ 1answer 65 views ### Inequality with a sum and factorial For a homework assignment we have the following question that I'm stuck on. Let$ 0 \leq y \leq 1 $be given.$\forall m \in \mathbb{N}$, define$ \displaystyle S_m(y)=\sum_{k=0}^m \binom{m}{k}y^k$. ... 2answers 403 views ### Induction step for$\sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$I want to prove by induction that,$\sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$OK I got the initial step, however, I have problems with the induction step: Here is what I tried: ... 2answers 163 views ### Fibonacci using proof by induction:$\sum_{i=1}^{n-2}F_i=F_n-2$everyone. I have been assigned an induction problem which requires me to use induction with the Fibonacci sequence. The summation states: $$\sum_{i=1}^{n-2}F_i=F_n-2\;,$$ with$F_0=F_1=1$. I ... 1answer 91 views ### Critique on a proof by induction that$\sum_{i=1}^n i^2= n(n+1)(2n+1)/6$? I need to make the proof for this 1:$$1^2 + 2^2 + 3^2 + ... + n^2=\frac{(n(n+1)(2n+1))}{6}$$ By mathematical induction I know that, If P(n) is true for$n>3^2then P(k) is also true for k=N and ... 1answer 47 views ### Inductive demonstration I have this: $$1^2+2^2+3^2+\dots+n^2=\frac{n(n+1)(2n+1)}6$$ So I was suggested of doing this: \begin{align} (1^2+2^2+3^2+\dots+k^2)+(k+1)^2 &= \frac{k(k+1)(2k+1)}6+(k+1)^2 \\ &= ... 3answers 399 views ### Summation and proof by induction question I can't figure this out based on examples in textbooks, etc. Show via induction that \sum_{j=1}^{n}j(j+1)(j+2)=\frac{n(n+1)(n+2)(n+3)}{4} So far, I have: (a) base case P(1)= 1(1+1)(1+2) = ... 2answers 394 views ### How to proof linearity property of summations with induction Recently I have faced with this question: {\sum_{k=1}^{n} (ca_k+ b_k) = c \sum_{k=1}^{n} a_k + \sum_{k=1}^{n} b_k } Proof linearity property of summations for all n ≥ 0 by using mathematical ... 3answers 648 views ### For the Fibonacci numbers, show for all n: F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1} The definition of a Fibonacci number is as follows:F_0=0\\F_1=1\\F_n=F_{n-1}+F_{n-2}\text{ for } n\geq 2$$Prove the given property of the Fibonacci numbers for all n greater than or equal to 1. ... 3answers 111 views ### Proving \sum_{k=1}^nk^3 = \left(\sum_{k=1}^n k\right)^2 using complete induction [duplicate] I tried to prove the following statement using complete induction but I couldn't manage to solve it because I got a complex notation eventually. The statement is the following:$$\sum_{k=1}^nk^3 = ... 3answers 483 views ### induction proof:\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}\$

I encountered the following induction proof on a practice exam for calculus: $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$ I have to prove this statement with induction. Can anyone please help me ...
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