AP EAMCET201920 Apr 2019Morning ShiftMathematicsSequences and SeriesActual
Assertion (A) ( (I) +(1+2+4)+(4+6+9) +(9+12+16)+ +(81+90+100)=1000 ) Reason (R) ( _ r=1 ^n (r^3-[r-1]^3 )=n^3 ) for any natural number (n )
Options
- ABoth ((A) ) and ((R) ) are true and ((R) ) is the correct explanation of ((A) )
- BBoth (A) and (R) are true but (R) is not the correct explanation of (A)
- C((A) ) is true but ((R) ) is false
- D(A) is false but (R) is true
Correct answer
A. Both ((A) ) and ((R) ) are true and ((R) ) is the correct explanation of ((A) )
Step-by-step solution
Since, ( aligned & 1+(1+2+4)+(4+6+9)+(9+12+16) & + +(81+90+100) & =1+ (1^2+(2 1)+2^2 )+ (2^2+(2 3)+3^2 )+ (3^2 . & .+(3 4)+4^2 )+ + (9^2+(9 10)+10^2 ) & = _ r=1 ¹⁰ [(r-1)^2+r(r-1)+r^2 ] & = _ r=1 ¹⁰[r-(r-1)] [(r-1)^2+r(r-1)+r^2 ] & = _ r=1 ¹⁰ [r^3-(r-1)^3 ] & = (1^3-0^3 )+ (2^3-1^3 )+ (3^3-2^3 )+ + (10^3-9^3 ) & =10^3-0^3=1000 aligned ) So, both (A) and (R) are true and (R) is the correct explanation of (A). Hence, option (1) is correct.