# Thread: Mathematical induction: sum of odd numbers.

1. ## Mathematical induction: sum of odd numbers.

by Set Theory PMI

Prove: for all Integers n greater than or = to 1.

1 + 3 + 5 + ... (2n - 1 ) = n^2

My initial solution but stuck as i went along the way...

------------------------------------------------------------------------------------------
i. when n = 1, then it is true
ii. if P (k) is true then P( k +1) is true...

im stuck then here please help i dont know what to do here nex. thanks a lot

1 + 3 + 5 +, ...(2n - 1) =n^2
then it is true

2. ## Re: the principle of mathematical induction... can anybody help me?

Base step: When n = 1, you have

LHS = 1

RHS = 1^2 = 1 = LHS.

Base step is proven.

Inductive step: Assume P(k) is true, i.e. that 1 + 3 + 5 + ... + (2k-1) = k^2. Using this you need to show that P(k+1) is true.

For P(k + 1) you are trying to show 1 + 3 + 5 + ... + (2k-1) + (2k+1) = (k + 1)^2.

LHS = 1 + 3 + 5 + ... + (2k - 1) + (2k + 1)

= k^2 + 2k + 1 since 1 + 3 + 5 + ... + (2k - 1) = k^2

= k^2 + k + k + 1

= k(k + 1) + 1(k + 1)

= (k + 1)(k + 1)

= (k + 1)^2

= RHS.

So the Inductive step is proven.

3. ## Re: the principle of mathematical induction... can anybody help me?

Originally Posted by rcs
by Set Theory PMI

Prove: for all Integers n greater than or = to 1.

1 + 3 + 5 + ... (2n - 1 ) = n^2

My initial solution but stuck as i went along the way...

------------------------------------------------------------------------------------------
i. when n = 1, then it is true
ii. if P (k) is true then P( k +1) is true...

im stuck then here please help i dont know what to do here nex. thanks a lot

1 + 3 + 5 +, ...(2n - 1) =n^2
then it is true
If you are stuck on the induction step,
then you might be wondering what you need to prove.

We want to show that the formula being true for "any" n
causes the formula to be true for the "next" n.

Let these successive values of n be termed "k" and "k+1".

Hence we have the following propositions

P(k)

$1+3+5+....+(2k-1)=k^2$

P(k+1)

$1+3+5+....+[2(k+1)-1]=(k+1)^2\;\;?$

We aim to show that IF P(k) is true, THEN P(k+1) is also true.

If P(k) is true, then P(k+1) becomes

$\left(1+3+5+.....+2k-1\right)+2k+1=k^2+2k+1\;\;?$

$\Rightarrow\ k^2+2k+1=k^2+2k+1$

Hence P(k+1) IS true, IF P(k) is true.

4. ## Thank you

thank you so much God BLess You are truly amazing.

5. ## RHS and LHS meant?

what does RHS and LHS meant for sir Prove It?

thanks

6. ## Re: RHS and LHS meant?

Originally Posted by rcs
what does RHS and LHS meant for sir Prove It?

thanks
Left Hand Side and Right Hand Side. This is standard notation.

7. ## Re: Mathematical induction: sum of odd numbers.

Originally Posted by rcs
by Set Theory PMI

Prove: for all Integers n greater than or = to 1.

1 + 3 + 5 + ... (2n - 1 ) = n^2

My initial solution but stuck as i went along the way...

------------------------------------------------------------------------------------------
i. when n = 1, then it is true
ii. if P (k) is true then P( k +1) is true...

im stuck then here please help i dont know what to do here nex. thanks a lot

1 + 3 + 5 +, ...(2n - 1) =n^2
then it is true
Without induction:

1 + 3 + 5 + 7 + 9 +... (2n-9)+(2n-7)+(2n-5)+(2n-3)+(2n - 1 )

Observe that:

1+(2n-1)=2n

1+(2n-1)=2n

3+(2n-3)=2n

5+(2n-5)=2n

7+(2n-7)=2n

.
.
.

The sum of all above is:

2n+2n+2n+2n+....

but how many times 2n appears?

8. ## Re: Mathematical induction: sum of odd numbers.

it is infinitely many

9. ## Re: Mathematical induction: sum of odd numbers.

\begin{aligned} & S = \sum_{1 \le k \le n}\left(2k-1\right) = \sum_{1 \le n-k \le n}\left[2(n-k)-1\right] = \sum_{1-n \le -k \le 0}\left(2n-2k-1\right) \\& = \sum_{0 \le k \le n-1}\left(2n-2k-1\right) = (n-1)-(2n-2n-1)+\sum_{1 \le k \le n}\left(2n-2k-1\right). \\& \begin{aligned}\therefore ~ 2S & = 2n+\sum_{1 \le k \le n}(2k-1+2n-2k-1) = 2n+\sum_{1 \le k \le n}(2n-2) \\& = 2n+(2n-2) \sum_{1 \le k \le n}(1) = 2n+2n(n-1) = 2n^2. \\& \therefore ~ S = n^2.\end{aligned} \end{aligned}

10. ## Re: Mathematical induction: sum of odd numbers.

Originally Posted by rcs
it is infinitely many
No! At the original sum there are a finite number of components so if we arrange them in pairs, we'll still have left with a finite number of components.

Try it with n=7, the sum of seven odd numbers:

1 + 3 + 5 + 7 + 9 + 11 + 13

1+13=14
3+11=14
5+9=14
and 7 left alone.

Try develop this theory of arithmetic summation by your own...

How To Be A Little Gauss

11. ## Re: Mathematical induction: sum of odd numbers.

Originally Posted by rcs
by Set Theory PMI

Prove: for all Integers n greater than or = to 1.

1 + 3 + 5 + ... (2n - 1 ) = n^2

My initial solution but stuck as i went along the way...

------------------------------------------------------------------------------------------
i. when n = 1, then it is true
ii. if P (k) is true then P( k +1) is true...

im stuck then here please help i dont know what to do here nex. thanks a lot

1 + 3 + 5 +, ...(2n - 1) =n^2
then it is true
check if it is true for n=1
1 = 1^2
1 = 1

Assume it is true for n=k
1 + 3 + 5 + ....(2k - 1) = k^2
Check if it is true for n= k + 1
1 + 3 + 5 + ....(2k -1) + (2(k+1) - 1) = (k+1)^2
You can replace 1 + 3 +5 +....(2k-1) with k^2
k^2 + (2(k+1) - 1) = (k+1)^2
k^2 + (2k +2 - 1) = (k+1)^2
k^2 + 2k + 1 = (k+1)^2
(k+1)(k+1) = (k+1)^2
(k+1)^2 = (k+1)^2

Proven!

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### using mathematical induction. the formula for sum of odd numbers

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