All Questions
6
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Induction proof for product of $a^x$ is less than or equal to the sum of $x\times a$
So this type of problem has me stuck in proving some relation. I assumed to use induction but I am stuck at a certain step and cannot understand if there is a trick or perhaps my idea is just wrong:
...
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2
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How to prove the following inequality with complete induction?
Let $n \in \mathbb{N}$, and let $a_1, ... , a_n > 0.$
Show that:
I got the hint that we have to use this induction step for the induction proof:
And thats what I got so far in the Induction step ...
- 33
2
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1
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Prove that $\sum_{{a_{n-1}}=1}^{a_n}\,\sum_{a_{n-2}=1}^{a_{n-1}}\,\ldots\,\sum_{a_{1}=1}^{a_2}\,a_1=\frac{\prod\limits_{i=0}^{n-1}\,(a_n+i)}{n!}$.
Use Principle of Mathematical Induction to show that, for every integer $n\ge2$,
$$\sum_{{a_{n-1}}=1}^{a_n}\,\sum_{a_{n-2}=1}^{a_{n-1}}\,\ldots\,\sum_{a_{1}=1}^{a_2}\,a_1=\frac{\prod\limits_{i=0}^{n-...
- 651
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How do you prove by induction when summation and product notation are involved?
Firstly, how would you solve an equation such as the Binomial Theorem by way of Mathematical Induction, and how could you use that to prove the following?
$$\left(1 + \frac1n\right)^n = 1 +\sum_{k=1}^...
user537153
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Inequality Induction Proof with Summation of Products
I'm doing some induction practice from the textbook Problems on Algorithms, and I couldn't figure out this problem:
For even $n \ge 4$ and $ 2 \le i \le \frac{n}{2}$:
$$
\sum_{k=1}^{i} \prod_{j=1}^{k}...
- 111
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2
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Proving there exists a set such that the sum of the elements equals the product
Show that for all odd positive integer $n$, there exists a set $A$ where $A= [a_1, a_2, a_3, ... , a_n]$ and $\displaystyle\sum_{i=1}^n a_i =\prod_{i=1}^n a_i$.
Edit: $a_1,...,a_n$ must be distinct.
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