We can assume that the first n-dimensional sphere and the subsequent
(n-k)-dimensional spheres are centered at the origin of the n-space by
translation.
x12 + x22 + ... + xk-12 + xk2 = rk2 (1)
and the n-dimensional sphere centered on it
x12 + x22 + ... + xk-12 + (xk - rk)2 = 1 (2)
where rn = 1.
Solving (1) and (2) gives
xk = rk - 1/(2*rk) (3)
Plugging xk into (1) or (2) gives
x12 + x22 + ... + xk-12 = 1 - 1/(4*rk2) (4)
Equation (4) implies the recurrence relation
rk-12 = 1 - 1/(4*rk2) (5)
I don't know if there is a procedure for finding a closed form solution
to a recurrence relation, but I came up with
rk2 = (n - k + 2)/[2*(n - k + 1)] (6)
rn2 = (n - n + 2)/[2*(n - n + 1)] = 1
r12 = (n - 1 + 2)/[2*(n - 1 + 1)] = (n + 1)/(2*n)
Therefore,
r1 = √[ (n + 1)/(2*n) ] (7)
QED
Check it out for n = 1, 2, and 3 and Charlie's post.
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