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Show that the integral part of (5 + ...

Show that the integral part of
` (5 + 2 sqrt(6))^(n) ` is odd where n is natural number

Text Solution

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`(5+2 sqrt(6))^(n) ` can be written as ` (5 + sqrt(24))^(n)`
Now , let
` I + f = (5 + sqrt(24))^(n)` …(i)
` 0 le f lt 1 `…(ii)
and let ` f' = (5 - sqrt(24))^(n)` …(iii)
` 0 lt f' lt 1` …(iv) ltbRgt On adding Eq.s (i) and (iii) , we get
` I + f + f' = (5 + sqrt(24)^(n)) + (5 - sqrt(24))^(n)`
` I = + 1 = 2p `
`AA p in ` N = Even integer [ from theorem 2]
` I = 2p - 1 = ` Odd integer
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