Class 11 Mathematics Sequence and Series: NCERT Solutions for Question 7

This page offers a step-by-step solution to the specific question NCERT Class 11th Mathematics - Sequence and Series | find the sum to nbsp n nbsp terms of the series 12 Answer from NCERT Class 11th Mathematics, Chapter Sequence and Series.
Question 7

Find the sum to n terms of the series 12 + (12 + 22) + (12 + 22 + 32) + …

Answer

The given series is 12 + (12 + 22) + (12 + 22 + 33 ) + …

an = (12 + 22 + 33 +…….+ n2)

equals fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction
equals fraction numerator n open parentheses 2 n squared space plus space 3 n space plus 1 close parentheses over denominator 6 end fraction space equals space fraction numerator 2 cubed space plus space 3 n squared space plus space n over denominator 6 end fraction
equals space 1 third n cubed space plus space 1 half space n squared space plus space 1 over 6 space n
therefore space S subscript n space equals space sum from k equals 1 to n of space a subscript k
space space space space space space space space space space equals sum from k equals 1 to n of space open parentheses 1 third space k cubed space plus space 1 half space k squared space plus space 1 over 6 space k close parentheses
space space space space space space space space space space equals space 1 third space sum from k equals 1 to n of space k cubed space plus space 1 half space sum from k equals 1 to n of space k squared space plus space 1 over 6 space sum from k equals 1 to n of space k
space space space space space space space space space equals 1 third fraction numerator n squared space open parentheses n plus 1 close parentheses squared over denominator open parentheses 2 close parentheses squared end fraction space plus space 1 half space cross times space fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction space plus space 1 over 6 space cross times fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction
space space space space space space space space space equals space fraction numerator n open parentheses n plus 1 close parentheses over denominator 6 end fraction space open square brackets fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction space plus space fraction numerator open parentheses 2 n plus 1 close parentheses over denominator 2 end fraction plus 1 half close square brackets
space space space space space space space space space equals space fraction numerator n open parentheses n plus 1 close parentheses over denominator 6 end fraction space open square brackets fraction numerator n squared space plus space n space plus space 2 n space plus 1 space plus 1 over denominator 2 end fraction close square brackets
space space space space space space space space equals space fraction numerator n open parentheses n plus 1 close parentheses over denominator 6 end fraction space open square brackets fraction numerator n open parentheses n plus 1 close parentheses plus 2 open parentheses n plus 1 close parentheses over denominator 2 end fraction close square brackets
space space space space space space space space equals space fraction numerator n open parentheses n plus 1 close parentheses over denominator 6 end fraction space open square brackets fraction numerator open parentheses n plus 1 close parentheses open parentheses n plus 2 close parentheses over denominator 2 end fraction close square brackets
space space space space space space space space equals space fraction numerator n open parentheses n plus 1 close parentheses squared open parentheses n plus 2 close parentheses over denominator 12 end fraction

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