Question
Find the sum of all odd numbers between:
$0$ and $50$
$0$ and $50$
✓
Answer
In this problem, we need to find the sum of all odd numbers lying between $0$ and $50.$
So, we know that the first odd number after $0$ is $1$ and the last odd number before $50$ is $49$.
Also. all these terms will form an A.P. with the common difference of $2$.
So here,
First term $(a) = 1$
Last term $(l) = 49$
Common difference $(d) = 2$
So, here the first step is to find the total number of terms. Let us take the number of terms as n.
Now, as we know,
$a_n = a + (n - 1)d$
So, for the last term,
$49 = 1 + (n - 1)2$
$49 = 1 + 2n - 2$
$49 = 2n - 1$
$49 + 1 = 2n$
Further simplifying,
$50 = 2n$
$\text{n}=\frac{50}{2}$
$n = 25$
Now, using the formula for the sum of n terms,
$\text{S}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$
For $n = 25$, we get
$\text{S}_{25}=\frac{25}{2}\Big[2\times1+(25-1)2\Big]$
$=\frac{25}{2}[2+48]$
$=\frac{25}{2}(50)=25\times25$
$=625$
Therefore, the sum of all the odd numberes lying between $0$ and $50$ is $S_n = 625.$
So, we know that the first odd number after $0$ is $1$ and the last odd number before $50$ is $49$.
Also. all these terms will form an A.P. with the common difference of $2$.
So here,
First term $(a) = 1$
Last term $(l) = 49$
Common difference $(d) = 2$
So, here the first step is to find the total number of terms. Let us take the number of terms as n.
Now, as we know,
$a_n = a + (n - 1)d$
So, for the last term,
$49 = 1 + (n - 1)2$
$49 = 1 + 2n - 2$
$49 = 2n - 1$
$49 + 1 = 2n$
Further simplifying,
$50 = 2n$
$\text{n}=\frac{50}{2}$
$n = 25$
Now, using the formula for the sum of n terms,
$\text{S}_\text{n}=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$
For $n = 25$, we get
$\text{S}_{25}=\frac{25}{2}\Big[2\times1+(25-1)2\Big]$
$=\frac{25}{2}[2+48]$
$=\frac{25}{2}(50)=25\times25$
$=625$
Therefore, the sum of all the odd numberes lying between $0$ and $50$ is $S_n = 625.$
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