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2-2-1-2-2-1-3-2-1-3-2-1-4-2-1-4-2-1-20-2-1-20-2-1-




Question Number 11838 by Peter last updated on 02/Apr/17
((2^2 +1)/(2^2 −1)) + ((3^2 +1)/(3^2 −1)) + ((4^2 +1)/(4^2 −1)) + .... + ((20^2 +1)/(20^2 −1)) = ....?
22+1221+32+1321+42+1421+.+202+12021=.?
Answered by ajfour last updated on 02/Apr/17
T_r = ((r^2 +1)/(r^2 −1)) = ((r^2 −1+2)/(r^2 −1))    =1+(2/((r−1)(r+1)))    T_r  = 1+(1/((r−1)))−(1/((r+1)))  S= Σ_(r=2) ^(r=20) T_r       = (1+(1/1)−(1/3))        +( 1+(1/2)−(1/4))        +(1+(1/3)−(1/5))        + (1+(1/4)−(1/6))         +....     ...+(1+(1/(17))−(1/(19)))         +(1+(1/(18))−(1/(20)))         +(1+(1/(19))−(1/(21)))  S = 19+1+(1/2)−(1/(20))−(1/(21))  S= 20+((169)/(420))  .
Tr=r2+1r21=r21+2r21=1+2(r1)(r+1)Tr=1+1(r1)1(r+1)S=r=20r=2Tr=(1+1113)+(1+1214)+(1+1315)+(1+1416)+.+(1+117119)+(1+118120)+(1+119121)S=19+1+12120121S=20+169420.

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