FIND FOUR NUMBERS IN AP WHOSE SUM IS 20 &SUM OF WHOSE SQUARES IS 120

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let four numbers in a.p. be:  

a-3d,a-d,a+d,a+3d

their sum=(a-3d)+(a-d)+

(a+d)+(a+3d)=20

=> 4a=20  =>a=5

also sum of squares=120

i.e.  (a-3d)2+(a-d)2

+(a+d)2+(a+3d)2=120 

  {where  2 means square }

=>a2-6ad+9d2-2ad +d2+

a2+2ad+d2+a2+6ad+9d2=120 

=>4a2=20d2=120 

=>4(25)=20d2=120

=>20d2+20  =>d2=1

=>d=+_1

hence the numbers are 

5-3,5-1,5-1,5-3

or  5-3,5-1,5-1,5-3

i.e.  2,4,6,8  or8,6,4,2

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