Show that the sum of all odd numbers lying between 10 and 100 is exactly three times the sum of odd numbers between 40 and 70  ????????/

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First find the sum of all odd numbers lying between 10 and 100.The first and last odd number between 10 and 100 are 11 and 99 respectively.So the corresponding series is,   11+13+15++99Here a=11, d=2, an=99This implies that,  a+n-1d=99 11+n-12=99   11+2n-2=99          2n+9=99               2n=99-9               2n=90                n=45So the required sum in this case is,   Sn=n2a+l, where l is last term   S45=45211+99   S45=452×110   S45=45×55   S45=2475   S45=3×825      .....1Now find the sum of all odd numbers lying between 40 and 70.The first and last odd number between 40 and 70 are 41 and 69 respectively.So the corresponding series is,   41+43+45++69Here a=41, d=2, an=69This implies that,  a+n-1d=69 41+n-12=69   41+2n-2=69          2n+39=69               2n=69-39               2n=30                n=15So the required sum in this case is,   Sn=n2a+l, where l is last term   S15=15241+69   S15=15×55   S15=825        .....2From 1  and 2, we get  sum of odd numbers between 10 and 100=3×sum of odd  numbers between 40 and 70

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