Question

If $1\times 2009+2\times 2008+3\times 2007+......+2009\times 1=2009\times 2011x$, then value of $x$ is

• A. $663$
• B. $670$
• C. $331$
• D. $335$

Answer 1

The correct option is D $335$

$2009\times 1+2\times 2008+...+2009\times 12(1\times 2009+2\times 2008+...1005\times 1005)=2(1\times (1005+1004)+2\times (1005+1003)+...1005\times 1005)=2(1\times 1005+1004+2\times 1005+2\times 1003+3\times 1005+3\times 1002+...1005\times 1005)=2((1\times 1005+2\times 1005+...1005\times 1005)+1004+2\times 1003+3\times 1002+1004)=2\left(1005(1+2+3+...1005)\right)+2(502\times (1+2+...502)+502+2\times 502+...)=2009\times 2011\times 335$

Hence$,$ $x=335.$