Question

Find the exact value of the series

Answer 1

Explanation:

`\frac{30\sqrt{2^(4n-2) +2^(4n+1) } }{(3^(n) +3^(n+1) )^2} \\=\frac{30\sqrt{2^(4n) *2^(-2) +2^(4n) *2^(1) } }{3^(2n) (1+3^(1)) } \\=\frac{30*2^(2n) \sqrt{(1)/(2^(2) )+2 } }{4*3^(2n) } \\=(30*2^(2n)*(3)/(2) )/(4*3^(2n) ) \\=(45(2^(2) )^(n) )/(4(3^(2) )^(n) ) \\=(45)/(4) *((4)/(9) )^(n) \\`

it is a G.P. with common ratio r=4/9 <1

`s=(a)/(1-r) ,a=first~term\\s=((45)/(4) *((16)/(81) ))/(1-(4)/(9) ) [by~n=2]\\s=(20)/(9) *(9)/(5) \\=4`