Divide both sides by 6^33:
(1/2)^33 + (2/3)^33 + (5/6)^33 < 1
Method 1:
First, take a look at a much smaller exponent.
Note that (1/2)^3 + (2/3)^3 + (5/6)^3 =
(1/8) + (8/27) + (125/216) =
27/216 + 64/216 + 125/216 = 216/216 = 1
So when the exponent is 3, the sum is exactly 1.
For any exponent larger than 3, for example 33, the sum is less than 1
(or equivalently note that 3^3 + 4^3 + 5^3 = 6^3)
Method 2:
(1/2)^33 + (2/3)^33 + (5/6)^33 < 1
will be true if the largest of the three terms is < 1/3
Is (5/6)^33 < 1/3 ?
Since (5/6)^7 = .27908... is already less than 1/3
raising to higher powers will be even smaller.
Edited on January 25, 2021, 7:51 am
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Posted by Larry
on 2021-01-25 07:46:36 |
this should do it
