explain full steps of this question 
1.3+3.5+5.7+........+ [2n-1][2n+1] is equal to n[4n^] + 6n-1 is divided by 3

Asked by Janvi Jhanga | 3rd Aug, 2014, 04:35: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space e q u a t i o n space 1 cross times 3 plus 3 cross times 5 plus 5 cross times 7 plus... plus open parentheses 2 n minus 1 close parentheses open parentheses 2 n plus 1 close parentheses equals fraction numerator n open square brackets 4 n squared plus 6 n minus 1 close square brackets over denominator 3 end fraction  A s s u m e space n equals 1 : T h u s space w e space h a v e comma L. H. S equals open parentheses 2 cross times 1 minus 1 close parentheses open parentheses 2 cross times 1 plus 1 close parentheses equals 3 R. H. S equals fraction numerator 1 open parentheses 4 cross times 1 squared plus 6 cross times 1 minus 1 close parentheses over denominator 3 end fraction equals 3 T h u s space t h e space s t a t e m e n t space i s space t r u e space f o r space n equals 1. L e t space u s space a s s u m e space t h a t space t h e space g i v e n space s t a t e m n e n t space i s space t r u e space f o r space n equals k T h u s comma space w e space h a v e comma space 1 cross times 3 plus 3 cross times 5 plus 5 cross times 7 plus... plus open parentheses 2 k minus 1 close parentheses open parentheses 2 k plus 1 close parentheses equals fraction numerator k open square brackets 4 k squared plus 6 k minus 1 close square brackets over denominator 3 end fraction W e space n e e d space t o space p r o v e space t h a t space t h e space g i v e n space s t a t e m e n t space i s space t r u e space f o r space n equals k plus 1. C o n s i d e r space t h e space L. H. S space a n d space p u t space n equals k plus 1. T h u s comma space w e space h a v e comma space 1 cross times 3 plus 3 cross times 5 plus 5 cross times 7 plus... plus open parentheses 2 k minus 1 close parentheses open parentheses 2 k plus 1 close parentheses plus open parentheses 2 open parentheses k plus 1 close parentheses minus 1 close parentheses open parentheses 2 open parentheses k plus 1 close parentheses plus 1 close parentheses equals fraction numerator k open square brackets 4 k squared plus 6 k minus 1 close square brackets over denominator 3 end fraction plus open parentheses 2 k plus 1 close parentheses open parentheses 2 k plus 3 close parentheses equals fraction numerator k open square brackets 4 k squared plus 6 k minus 1 close square brackets over denominator 3 end fraction plus open square brackets 4 k squared plus 8 k plus 3 close square brackets equals fraction numerator k open square brackets 4 k squared plus 6 k minus 1 close square brackets plus 3 open square brackets 4 k squared plus 8 k plus 3 close square brackets over denominator 3 end fraction equals fraction numerator open square brackets 4 k cubed plus 6 k squared minus k close square brackets plus open square brackets 12 k squared plus 24 k plus 9 close square brackets over denominator 3 end fraction equals fraction numerator open square brackets 4 k cubed plus 18 k squared plus 23 k plus 9 close square brackets over denominator 3 end fraction equals fraction numerator open parentheses k plus 1 close parentheses open square brackets 4 k squared plus 14 k plus 9 close square brackets over denominator 3 end fraction equals fraction numerator open parentheses k plus 1 close parentheses open square brackets 4 k squared plus 8 k plus 4 plus 6 k plus 6 minus 1 close square brackets over denominator 3 end fraction equals fraction numerator open parentheses k plus 1 close parentheses open square brackets 4 open parentheses k squared plus 2 k plus 1 close parentheses plus 6 open parentheses k plus 1 close parentheses minus 1 close square brackets over denominator 3 end fraction equals fraction numerator open parentheses k plus 1 close parentheses open square brackets 4 open parentheses k plus 1 close parentheses squared plus 6 open parentheses k plus 1 close parentheses minus 1 close square brackets over denominator 3 end fraction T h u s comma space t h e space g i v e n space s t a t e m e n t space i s space t r u e space f o r space n equals k plus 1 T h u s space t h e space g i v e n space s t a t e m e n t space i s space t r u e space f o r space n.

Answered by Vimala Ramamurthy | 4th Aug, 2014, 09:14: AM