Let = 1 4 + 1 8 + 1 16 + and = 1 3 + 1 9 + 1 27 + . Then the value of (0.2)^ _ 5 ( ) +(0.04)^ ₅( ) is equal to:

Step-by-step solution

The sum of an infinite geometric progression is given by S_ = a 1-r . For , a = 1 4 and r = 1 2 : = 1 4 1 - 1 2 = 1 2 For , a = 1 3 and r = 1 3 : = 1 3 1 - 1 3 = 1 2 Now, evaluating the first term: (0.2)^ _ 5 ( ) = (5⁻¹ )^ _ 5^ 1/2 (1/2) Using the property _ a^k (x) = 1 k _a(x) : _ 5^ 1/2 (1/2) = 2 ₅(1/2) = ₅(1/4) Thus, (5⁻¹ )^ ₅(1/4) = 5^ - ₅(1/4) = 5^ ₅(4) = 4 Evaluating the second term: (0.04)^ ₅( ) = (5⁻² )^ ₅(1/2) (5⁻² )^ ₅(1/2) = 5^ -2 ₅(1/2) = 5^ ₅(4) = 4 Adding both terms: (0.2)^ _ 5 ( ) + (0.04)^ ₅( ) = 4