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

Question:
Find the sum to n terms of the series whose nth terms is given by n² + 2n

Answer:

a_n = n² + 2n

$therefore space S subscript n space equals space sum from k equals 1 to n of space k squared space plus space 2 to the power of k space equals space sum from k equals 1 to n of space k squared space space plus space sum from k equals 1 to n of space 2 to the power of k space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis C o n s i d e r space sum from k equals 1 to n of space 2 to the power of k space equals space 2 to the power of 1 space plus space 2 squared space plus 2 cubed space plus space...$

The above series 2, 2², 2³, … is a G.P. with both the first term and common ratio equal to 2.

$therefore space sum from k equals 1 to n of space 2 to the power of k space equals space fraction numerator open parentheses 2 close parentheses open square brackets open parentheses 2 close parentheses to the power of n space minus 1 close square brackets over denominator 2 minus 1 end fraction equals 2 open parentheses 2 to the power of n space minus 1 close parentheses space space space space space space space space space space space... left parenthesis 2 right parenthesis$

Therefore, from (1) and (2), we obtain

$S subscript n space equals space sum from k equals 1 to n of space k squared space plus space 2 open parentheses 2 to the power of n space minus 1 close parentheses space equals space fraction numerator n open parentheses n plus 1 close parentheses open parentheses 2 n plus 1 close parentheses over denominator 6 end fraction plus 2 open parentheses 2 to the power of n space end exponent minus 1 close parentheses$

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Question:
Find the sum to n terms of the series whose nth terms is given by n² + 2n

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Question: Find the sum to n terms of the series whose nth terms is given by n2 + 2n

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Question:
Find the sum to n terms of the series whose nth terms is given by n² + 2n