f(n) = (1 + √2)^n = √x + √(x+1)

 
 
f(n) \; = \; ( \, 1 + \sqrt{2} \, )^{ \,n}
 

( \, 1 + \sqrt{2} \, )^1 \; = \; \sqrt{ \, 1 \, } \; + \; \sqrt{ \, 1 + 1 \, }

( \, 1 + \sqrt{2} \, )^2 \; = \; \sqrt{ \, 8 \, } \; + \; \sqrt{ \, 8 + 1 \, }

( \, 1 + \sqrt{2} \, )^3 \; = \; \sqrt{ \, 49 \, } \; + \; \sqrt{ \, 49 + 1 \, }

( \, 1 + \sqrt{2} \, )^4 \; = \; \sqrt{ \, 288 \, } \; + \; \sqrt{ \, 288 + 1 \, }

( \, 1 + \sqrt{2} \, )^5 \; = \; \sqrt{ \, 1681 \, } \; + \; \sqrt{1681 + 1 \, }

( \, 1 + \sqrt{2} \, )^6 \; = \; \sqrt{ \, 9800 \, } \; + \; \sqrt{9800 + 1 \, }

( \, 1 + \sqrt{2} \, )^7 \; = \; \sqrt{ \, 57121 \, } \; + \; \sqrt{57121 + 1 \, }

( \, 1 + \sqrt{2} \, )^8 \; = \; \sqrt{ \, 332928 \, } \; + \; \sqrt{332928 + 1 \, }

( \, 1 + \sqrt{2} \, )^9 \; = \; \sqrt{ \, 1940449 \, } \; + \; \sqrt{1940449 + 1 \, }

( \, 1 + \sqrt{2} \, )^{10} \; = \; \sqrt{ \, 11309768 \, } \; + \; \sqrt{11309768 + 1 \, }

 
Show every positive integer   n > 10,   f(n)   can be expressed as a sum of square roots of consecutive integers
 

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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