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=93⋮Sn=2n+13.
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