Question

Find the sum of all the odd numbers between 100 and 200.

Answer 1

Hint – In this particular type of question use the concept that odd number is not divisible by the even number so first find out, first, second and the last odd number between 100 and 200 then check which series these number formed then apply the ${n^{th}}$ term formula of the series, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
We have to find the sum of all the odd numbers between 100 and 200.
As we know that the odd numbers are those who do not divide by 2 or by an even number.
So the set of odd numbers between 100 and 200 are
101, 103, 105,..........................., 199
Now as we know (101, 103, 105,..........................., 199) forms an A.P with first term (a = 101), common difference (d = (103 - 101) = (105 - 103) = 2), last term ${a_n} = 199$ and the number of terms is n.
Now as we know that the last term of an A.P is given as
$ \Rightarrow {a_n} = a + \left( {n - 1} \right)d$ Where, symbols have their usual meaning.
Now substitute the value in the above equation we have,
$ \Rightarrow 199 = 101 + \left( {n - 1} \right)2$
Now simplify this and calculate the number of terms in this A.P we have,
$ \Rightarrow 199 - 101 = \left( {n - 1} \right)2$
$ \Rightarrow \dfrac{{98}}{2} = \left( {n - 1} \right)$
$ \Rightarrow 49 + 1 = n$
$ \Rightarrow n = 50$
Now we know the formula of sum of an A.P which is given as
${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$, Where (a) is first term, (d) is common difference and (n) is number of terms.
So the sum of all the odd numbers between 100 and 200 is
$ \Rightarrow {S_{odd}} = \dfrac{{50}}{2}\left( {2\left( {101} \right) + \left( {50 - 1} \right)2} \right)$
Now simplify the above equation we have,
$ \Rightarrow {S_{odd}} = 25\left( {202 + 98} \right) = 25\left( {300} \right) = 7500$
So this is the required sum of all the odd numbers between 100 and 200.
So this is the required answer.

Note – Whenever we face such types of questions the key concept we have to remember that the ${n^{th}}$ term formula of the A.P is given as ${a_n} = a + \left( {n - 1} \right)d$, where symbols have their usual meanings so substitute all the values in this equation and calculate the number of terms in the series, we will get the required answer.