SOLUTION: Simplify: {{{(1/4) + (1/4^2) + ((4^2)/(4^3)) + ((4^2)/(4^4)) + ((4^4)/(4^5)) + ((4^4)/(4^6))}}} + ... + {{{((4^102)/(4^103)) + ((4^102)/(4^104))}}}

Algebra ->  Exponents -> SOLUTION: Simplify: {{{(1/4) + (1/4^2) + ((4^2)/(4^3)) + ((4^2)/(4^4)) + ((4^4)/(4^5)) + ((4^4)/(4^6))}}} + ... + {{{((4^102)/(4^103)) + ((4^102)/(4^104))}}}      Log On


   

Question 1199206: Simplify: + ... + %28%284%5E102%29%2F%284%5E103%29%29+%2B+%28%284%5E102%29%2F%284%5E104%29%29
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

+ ... + %28%284%5E102%29%2F%284%5E103%29%29+%2B+%28%284%5E102%29%2F%284%5E104%29%29

1%2F4+%2B+1%2F16+%2B+1%2F4+%2B+1%2F16%2B+1%2F4%2B+1%2F16 + ... + 1%2F4+%2B+1%2F16
15%2F16 + ... + 5%2F16

Answer by ikleyn(52818) About Me  (Show Source):
You can put this solution on YOUR website!
.
Simplify: + ... + %28%284%5E102%29%2F%284%5E103%29%29+%2B+%28%284%5E102%29%2F%284%5E104%29%29
~~~~~~~~~~~~~~~~~~~~~~~~~ 

I see two sequences alternate.


After separating, one sequence is   4%5E0%2F4%5E1 + 4%5E2%2F4%5E3 + 4%5E4%2F4%5E5 + . . . + 4%5E102%2F4%5E103         (1)


              Another sequence is   4%5E0%2F4%5E2 + 4%5E2%2F4%5E4 + 4%5E6%2F4%5E8 + . . . + 4%5E102%2F4%5E104         (2)


Sequence (1), after reducing, is the sum of 102%2F2%2B1 = 52 equal addends, each of which is  1%2F4,
so this sum is  52%2F4.


Sequence (2), after reducing, is the sum of 102%2F2%2B1 = 52 equal addends, each of which is  1%2F16,
so this sum is  52%2F16.


Therefore, the total sum is  52%2F4 + 52%2F16 = %2852%2A4%29%2F16 + 52%2F16 = 260%2F16 = 130%2F8 = 162%2F8 = 161%2F4.    ANSWER


Solved.