Find the value of _ 5 ( (5 5 +5) (5 5 +5)^2 (5 5 +5)^3 (6+2 5 ) [3] 215-18 [3] 215-18 [3] 215-18 [3] )

Step-by-step solution

Using AP-GP (5 5 +5) (5 5 +5)^2 (5 5 +5)^3 =(5 5 +5)^ 1 2 + 2 4 + 3 8 + =(5( 5 +1))^2 Let 1 2 + 2 4 + 3 8 + =2 aligned S & = 1 2 + 2 4 + 3 8 + (1) 2 S & = 1 1 + 2 2 + 3 4 + (2) aligned S=1+ 1 2 + 1 4 + 1 8 + . [subtracting Eqn. (1) from Eqn. (2)] Now geometric progression, with a=1 and r= 1 2 , and thus evaluates to 1 1-(1 / 2) =2 . Now note that 6+2 5 =( 5 +1)^2 As a last step, recall the identity aligned (a-b)^3 & =a^3-b^3-3 a b(a-b) (a-b) & = [3] a^3-b^3-3 a b(a-b) & = [3] a^3-b^3-3 a b [3] a^3-b^3-3 a b(a-b) &