Sequences and Series

If 8=3+ 14(3+p)+ 142(3+2p)+ 143(3+3p)+ , then the value of p is .

Answer

9

Solution

RHS = r=0 3+rp4r=3 r=0 14r+p r=1 r4r

∵ r=0 xr= 11-x and r=1 r xr= x(1-x)2 with x= 14

=3 43+p 1/4(3/4)2=4+ 4p9

8=4+ 4p9 ⇒ p=9

Answer: 9