Subjective Type

Find the sum of odd integers from $$1$$ to $$2001$$.

Solution

The odd integers from$$1$$ to $$2001$$ are $$1, 3, 5, ...1999, 2001$$.

This sequence form an A.P.

Here, first term is, $$a= 1$$ and common difference, $$d=2$$

Here $$
a+(n-1)d=2001 $$

$$\Rightarrow 1+(n-1)(2)=2001$$

$$\Rightarrow 2n-2=2000$$

$$\Rightarrow n=1001$$

Hence required sum is,

$$\displaystyle { S }_{ n }=\frac { n }{ 2 } \left[ 2a+(n-1)d

\right] $$

$$=\dfrac { 1001 }{ 2 } \left[ 2\times 1+(1001-1)\times 2 \right]$$

$$\displaystyle = \dfrac { 1001 }{ 2 } \left[ 2+1000\times 2 \right] $$

$$= \dfrac { 1001 }{ 2 } \times 2002 $$

$$= 1001\times 1001 $$

$$=1002001$$

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