Sum of the series to n terms:
2.1+3.2+4.4+5.8+.....

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Please find below the solution to the asked query:

S = 2.1+3.2+4.4+5.8+.........+Tn2,3,4....is A.P. while 1,2,4....is an G.P.nth term of an A.P. is a+n-1d and that of G.P. is arn-1Here a=2, d=1 and r=2Tn of given series is 2+n-11.1.2n-1Tn=n+1.2n-1Again S        =    2.1+3.2+4.4+5.8+.........+Tn 2S  =            2.2+3.4+4.8+...........+Tn-1+TnSubtracting above two equations we get-S=2+2+4+8+...........+Tn-Tn-1-Tn ; equationiAgain 2+4+8+...........+Tn-Tn-1 is a G.P. of n-1 terms.Sum=arn-1-1r-1=22n-1-12-1=2.2n-1-2Sum =2n-2Hence equationi becomes,-S=2+2n-2-Tn-S=2n-TnS=Tn-2nPutting the value of Tn we get,S=n+1.2n-1-2n=2n-1n+1-2S=2n-1n-1Hence sum of given series is 2n-1n-1

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