Difference Of Nested Radicals
S 1 S 2 = = n + n − n + n − ⋯ n − n + n − n + ⋯
When both infinitely nested radicals are defined, what is S 1 − S 2 ?
The answer is 1.
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3 solutions
For small n ( n = 1 . 1 , say) these nested radicals are not even defined.
ya for what if we have n 254
Basically:
S 1 2 = n + S 2
S 2 2 = n − S 1
Subtracting the 2 equations give:
S 1 2 − S 2 2 = S 1 + S 2
Factor the LHS as difference of 2 squares and dividing will yield the answer.
Use quadratic equation to prove
S 1 = n + S 2 and S 2 = n − S 1 Squaring both equations: S 1 2 = n + S 2 and S 2 2 = n − S 1
Subtract them: S 1 2 − S 2 2 = S 2 + S 1 → ( S 1 + S 2 ) ( S 1 − S 2 ) = S 1 + S 2
Since S 1 + S 2 = 0 → S 1 − S 2 = 1