Wanna practice Induction ?

The principle of mathematical induction is a very very useful tool in many proofs, and also in proving some useful formulas. $\color{Red}{\text{The principle states that any given set of positive integers}}$ has ALL natural numbers if it follows the following conditions -

$(i)$ The first natural number, i.e. $1$ is in the set.

$(ii)$ Whenever the integer $k$ is in the set, then $k+1$ is also in the set.

This looks so obvious, see that if you have been given the two statements simultaneously, then by using $(ii)$ on $(i)$ , you can say that $2$ is in the set. Then again applying the statement $(ii)$ for $2$ , we say that $(3)$ will be in the set, so on.

$\color{#3D99F6}{\textbf{Thus if you prove a certain result for 1}}$ , $\color{#3D99F6}{\textbf{and then assume that the result is true}}$ for $k$ , just prove that it is true for $k+1$ and you're done !

Same thing you can do for 0, then prove for whole numbers !!!

Problems for practice:-

Prove that the following results hold $\forall n \in \mathbb{N}$

$(a) \displaystyle \sum_{k=0} ^n 2^k = 2^{n+1}-1$

$(b) \displaystyle \sum_{k=1} ^n k = \dfrac{n(n+1)}{2}$

$(c) \displaystyle \sum_{k=1} ^n k^2 = \dfrac{n(n+1)(2n+1)}{6}$

$(d) \displaystyle \sum_{k=1}^n k^3 = \biggl( \sum_{k=0} ^n k \biggr) ^2$

$(e) \displaystyle 24 \mid 2\cdot 7^n + 3\cdot 5^n -5$

$(f) \displaystyle 1\cdot 2+2\cdot 3+3\cdot 4+ .... + n(n+1) = \dfrac{n(n+1)(n+2)}{3}$

I felt like sharing because though it's very well known to all, I am willing to find some good level problems for practicing this foundation builder concept once more... $\color{#20A900}{\text{Isn't this a good revision ?}}$

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2^{34}

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\sqrt{2}

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Aditya Raut this is really a nice note for practice...... could you please add more divisibility kinda problems to give a more diverse experience to readers? ........ thanks :)

Abhinav Raichur - 6 years, 11 months ago

Here is some additional problems I found on the internet: http://staffhome.ecm.uwa.edu.au/~00021149/Academy/1995/inductionprobs.pdf

Daniel Liu - 6 years, 11 months ago

Thanks - they look like a good selection of induction problems

Curtis Clement - 6 years, 5 months ago

I actually haven't tried $(d)$ and $(f)$ before since doing math XD

Here's the solution for $(e)$ if someone got stuck.

$2\times 7^{m+1} + 3 \times 5^{m+1} - 5$

$= 2\times 7^{m}\times 7 + 3\times 5^{m}\times 5 - 5$

$= 14\times 7^{m} + 15 \times 5^{m} - 5$

$= 12\times 7^{m} + 12\times 5^{m} + 2\times 7^{m} + 3\times 5^{m} - 5$

$= 12(7^{m} + 5^{m}) + 24k$ for some $k$

$= 24n + 24k$ for some $n$

Samuraiwarm Tsunayoshi - 6 years, 11 months ago

$(a)$ Base Case $n = 1$

$\displaystyle \sum_{k = 0}^{1}2^{k} = 3$

$2^{n+1} - 1 = 3$

True.

Propose it is true $\forall n \in N$

Then it must be true for $n + 1$

$\displaystyle\sum_{k = 0}^{n+1}2^{k} = \sum_{k = 0}^{n}2^{k} + 2^{n+1}$

Substituting the value from our proposal,

$\sum_{k = 0}^{n}2^{k} + 2^{n+1} = 2^{n+1} - 1 + 2^{n+1}= 2^{n+2} - 1$

Thus, $LHS = RHS$

Thus, $LHS = RHS \forall n \in N$

Sorry for ugly arrangement.

Sanchayapol Lewgasamsarn - 6 years, 10 months ago

@Aditya Raut Can you add these to the Induction Wiki page? Thanks!

Calvin Lin Staff - 6 years, 8 months ago

Nice! I'm too into creating notes and problems set for inequality :) Why not take a look at it @Aditya Raut ? :D

Priyansh Sangule - 6 years, 10 months ago

What is e) ?... in d) second summation is it k=0 or k=1 ? Nice collection.

Niranjan Khanderia - 6 years, 8 months ago

Thanks, but Just think ! Will adding $0$ make any change? In that summation, starting from $k=0$ will yield same thing as starting with $k=1$ , won't it ? @Niranjan Khanderia

Aditya Raut - 6 years, 7 months ago

Thanks.

Niranjan Khanderia - 6 years, 7 months ago

And e) means prove that for all positive integers $n$ , $(2\cdot 7^n + 3\cdot 5^n -5)$ is divisible by $24$ .

$a\mid b$ is the symbol of "a divides b" or "b is divisible by a".

Latex code for the symbol $\mid$ is "\mid"

Thanks. I did not get it since I did not assume the parenthesis.

/(good/)

samuel ayinde - 6 years, 5 months ago

A good level problem is here: Fro every positive integer $n$ , prove that $\sqrt{(4n+1)}<\sqrt n +\sqrt{(n+1)}<\sqrt{(4n+2)}$ . Hence or otherwise, prove that [ $\sqrt{n}+\sqrt{n+1}$ ]=[ $\sqrt{4n+1}$ ] where [.] denotes floor function. It is an IITJEE problem.

Gautam Sharma - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$