Let `a_n= 16,4,1,…` be a geometric sequence .Define `P_n` as the product of the first n terms. The value of `Sigma_(n=1)^(oo) nsqrtP_n` is _________.

Answer

32

Explanation

For the G.P. `a,ar,ar^(2),…`
`P_(n)=a(ar)(ar^(2))…(ar^(n-1))=a^(n)r^(n(n-1)//2)`
`thereforesum_(n=1)^(oo)rootn(P_(n))=sum_(n=1)^(oo)ar^((n-1)//2)`
Now, `sum_(n=1)^(oo)ar^((n-1)//2)=a[1+sqrtr+r+rsqrtr+..oo]=a/(1-sqrtr)`
Given, a=16 and r=1/4
`thereforeS=16/(1-(1//2))=32`