Find the sum of n terms of the series the rth term of which is (2r+1)2^r - Maths - Sequences and Series

Question

Find the sum of n terms of the series the rth term of which is
(2r+1)2^r

Priyanka Kedia · Accepted Answer

Given, the rth term of a series Tr=2r+12rSo,T1=321T2=522T3=723::Tn=2n+12nThus,Sn=T1+T2+T3+
+Tn-1+TnSn=321+522+723+
+2n-12n-1+2n+12n2Sn=       +322+523+724+    
   +2n-12n+2n+12n+1Subtracting above two equations, we have1-2Sn=321+222+223+224+    
   +22n+2n+12n+1⇒-Sn=6+222+23+24+    
   +2n+2n+12n+1⇒-Sn=6+2222n-1-12-1+2n+12n+1⇒-Sn=6+2n+2-8+n2n+2+2n+1⇒-Sn=2n+2+12n+1-2⇒Sn=2-2n+32n+1