Friday, 20 September 2013

summation - Combination of quadratic and arithmetic series



Problem:




Calculate 12+22+32+42++2333333021+2+3+4++23333330.








Attempt:



I know the denominator is arithmetic series and equals
n2(T1+Tn)=233333302(1+23333330)=272222156111115,


but how do I calculate the numerator without using a calculator?


Answer



Intuitively,

S1=121=1=33S2=12+221+2=53S3=12+22+321+2+3=73S4=12+22+32+421+2+3+4=3=93Sn=2n+13.


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