Question

Find the sum of odd integers from 1 and 2001.

Answer 1

Hint – In this question write the series of odd integers starting from one till 2001, that is 1, 3, 5, 7……………. 2001. This series forms an arithmetic progression so take out the common difference, and the number of terms in this series. Then use the direct formula for sum of n terms of a series that is ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$. This will help getting the answer.

Complete step-by-step answer:
As we know that the odd integers are not divided by 2.
So the set of odd integers $\left\{ {1,3,5,7...........} \right\}$
Now we have to find the sum of odd integers from 1 and 2001.
So the set of odd integers from 1 and 2001 is
1, 3, 5, 7... 2001
Now as we see this follows the rule arithmetic progression (A.P), with first term $\left( {{a_1} = 1} \right)$, common difference (d) = $\left( {3 - 1} \right) = \left( {5 - 3} \right) = 2$ and the last term $\left( {{a_n} = 2001} \right)$.
Now as we know that the ${n^{th}}$ of an A.P is given as
$ \Rightarrow {a_n} = {a_1} + \left( {n - 1} \right)d$, where symbols have their usual meanings and n = number of terms.
Now substitute the values we have,
$ \Rightarrow 2001 = 1 + \left( {n - 1} \right)2$
Now simplify this we have,
$ \Rightarrow 2001 - 1 = 2\left( {n - 1} \right)$
$ \Rightarrow n - 1 = \dfrac{{2000}}{2} = 1000$
$ \Rightarrow n = 1000 + 1 = 1001$
So the number of terms in the series is 1001.
Now we have to find the sum
1 + 3 + 5 + 7 +.............. + 2001
And the formula is given as
${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$, where symbols have their usual meaning.
Now substitute the values we have,
$ \Rightarrow {S_n} = \dfrac{{1001}}{2}\left( {2 \times 1 + \left( {1001 - 1} \right)2} \right)$
$ \Rightarrow {S_n} = 1001\left( {1 + 1000} \right)$
$ \Rightarrow {S_n} = 1001\left( {1001} \right) = 1002001$
So this is the required sum of odd integers from 1 and 2001.

Note – An odd number is one which is exactly non-divisible by 2. That is the remainder of the number when divided by 2 is not zero. A series is said to be in A.P if and only if the common difference that is the difference between the consecutive numbers of the series remains constant throughout. It is advised to remember some basic series formulas as it helps solving problems of this kind.