Question
Find the sum of the odd numbers between $0$ and $50.$
✓
Answer
The odd numbers between $0$ and $50$ are $1, 3, 5, 7, ....., 49.$
$\text { Here, } a_2-a_4=3-1=2$
$ a_3-a_2=5-3=2 $
$a_4-a_3=7-5=2$
i.e. $a_{k+1}- a_k$ is the same everytime.
So, the above list of numbers forms an $AP.$
Here, $a = 1$
$d = 2$
$l = 49$
Let the number of terms of the $AP$ be $n.$
Then,$ l = a + (n - 1)d$
$ \Rightarrow 49 = 1 + (n - 1)d$
$ \Rightarrow (n - 1)2 = 48 $
$ \Rightarrow n - 1 = \frac{{48}}{2}$
$ \Rightarrow n - 1 = 24$
$ \Rightarrow n = 24 + 1$
$ \Rightarrow n = 25$
Hence, the number of terms of the $AP$ be $25.$
$\therefore $ Sum of the odd numbers between 0 and $50 = S_{25}$
$ = \frac{{25}}{2}(a + l)$ ......$\because {S_n} = \frac{n}{2}(a + l)$
$ = \left( {\frac{{25}}{2}} \right)(1 + 49)$
$ = \left( {\frac{{25}}{2}} \right)(50)$
$= (25) (25)$
$= 625$
$\text { Here, } a_2-a_4=3-1=2$
$ a_3-a_2=5-3=2 $
$a_4-a_3=7-5=2$
i.e. $a_{k+1}- a_k$ is the same everytime.
So, the above list of numbers forms an $AP.$
Here, $a = 1$
$d = 2$
$l = 49$
Let the number of terms of the $AP$ be $n.$
Then,$ l = a + (n - 1)d$
$ \Rightarrow 49 = 1 + (n - 1)d$
$ \Rightarrow (n - 1)2 = 48 $
$ \Rightarrow n - 1 = \frac{{48}}{2}$
$ \Rightarrow n - 1 = 24$
$ \Rightarrow n = 24 + 1$
$ \Rightarrow n = 25$
Hence, the number of terms of the $AP$ be $25.$
$\therefore $ Sum of the odd numbers between 0 and $50 = S_{25}$
$ = \frac{{25}}{2}(a + l)$ ......$\because {S_n} = \frac{n}{2}(a + l)$
$ = \left( {\frac{{25}}{2}} \right)(1 + 49)$
$ = \left( {\frac{{25}}{2}} \right)(50)$
$= (25) (25)$
$= 625$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Show that the following numbers are irrational.
$3-\sqrt{5}$
→$3-\sqrt{5}$
$200$ logs are stacked in the following manner: $20$ logs in the bottom row, $19$ in the next row, $18$ in the row next to it and so on (see Fig.). In how many rows are the $200$ logs placed and how many logs are in the top row?
→A solid metallic sphere of diameter $8\ cm$ is melted and drawn into a cylindrical wire of uniform width. If the length of the wire is $12m,$ then find its width.
→If $1$ is a root of the quadratic equation $3 x^2+a x-2=0 $ and the quadratic equation $a(x^2+ 6x) - b = 0$ has equal roots, find the value of $b.$
→Find the volume of a solid in the form of a right circular cylinder with hemi-spherical ends whose total length is $2.7m$ and the diameter of each hemi-spherical end is $0.7m.$
→Show that the sequence defined by $a_n=3 n^2-5$ is not an $A.P.$
→Ramkali would need $Rs. 1800$ for admission fee and books etc., for her daughter to start going to school from next year. She saved $Rs. 50$ in the first month of this year and increased her monthly saving by $Rs. 20.$ After a year, how much money will she save$?$ Will she be able to fulfil her dream of sending her daughter to school$?$
→Solve the following quadratic equation:
$ x^2-(2 b-1) x+\left(b^2-b-20\right)=0 $
→$ x^2-(2 b-1) x+\left(b^2-b-20\right)=0 $
In an isosceles triangle $ABC, AB = AC = 25\ cm, BC = 14\ cm,$ Calculate the altitude from $A$ on $BC.$
→Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.