NCERT Solution
Class 11
Mathematics
Sequence and Series
find the sum to nbsp n nbsp terms of the series 12...

Question:
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 ) + …

a_n = (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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Question:
Find the sum to n terms of the series 1² + (1² + 2²) + (1² + 2² + 3²) + …

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

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