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The sum of the first n odd natural numbers: $1+3+5+7+...+(2n-1)$.

My question is: Why do we write the last element as $2n-1$ ? Why don't we write $1+3+5+7+...+(2n+1)$ ?

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    $2$ is not odd! – 2017-01-21
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    @Kahler Do you know how to find the sum? Or are you only asking why we write $2n-1$ instead of $2n+1$? – 2017-01-21
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    @projectilemotion Yes. It is equal to $n^2$ by the induction on $n$. – 2017-01-21

4 Answers 4

1

In the expression $2n-1$, the value goes from $1 \to n$ for the values for $n$ to get $1,3,5 ... 2n-1$

Since $n$ is defined over $\mathbb N$.

In the expression $2n+1$, if you want to keep the values as in the previous expression you would need to go from $0$ to $n-1$, for the values of $n$.

But that's not how it works by its definition.

Since $n$ is defined over $\mathbb N$, the second expression gives $3,5,7...2n+1$

These two are not the same since $n\in \mathbb N$ always follows natural values from $1$ to $n$; $(1,2,3,...n)$

The second expression can be still used instead of the first one, but only if you say beforehand that $n\in \mathbb W$ and that the value $n$ itself is excluded, leaving the $n-1$ the last value.

But it really makes no sense to do that. You would only make things inconvenient.

W stands for whole numbers, and N stands for natural numbers.

$\mathbb N = \{1,2,3... \}$

$\mathbb W = \{0,1,2,3... \}$

6

That is a simple matter of counting. The first natural odd number is

$$2\cdot(1) - 1 = 1$$

The second is

$$2\cdot(2) - 1 = 3$$

The third is

$$2\cdot(3) - 1 = 5$$

...

The 57th is

$$2\cdot(57) - 1 = 113$$

And the $n $th is

$$2\cdot(n) - 1 = 2n - 1$$

So the first $n $ odd natural numbers end with $(2n-1) $.

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    You're answers are nicely formatted :-) – 2017-01-21
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    @SimplyBeautifulArt thanks :D – 2017-01-21
4

We write $2n-1$ as we are using $n$ terms in the expression. The 1st term is 1, the 2nd term is 3, the 3rd term is 5, and so on until the nth term is $2n-1$.

This can be seen by looking at the numbers and thinking about what formula gives that pattern. E.g.

$1\rightarrow1$

$2\rightarrow3$

$3\rightarrow5$

$\cdots$

$n\rightarrow?$

1

I'ts for counting set $\mathbb{N}$ that is used for indexing terms of sequence or series. The $n$-th term of a sequence is $a_n$ and first term generally is began with index $1$, the first member of $\mathbb{N}$.