Given that x is a nonzero real number.
Determine the possible value(s) of x satisfying this equation:
(1+1/x)x+1 = (1+1/7)7
*** Adapted from a problem appearing in an Australian Mathematics Competition.
clearvars,clc
for x=-12:.1:-7
disp([x (1+1/x)^(x+1)-(1+1/7)^7])
end
fplot(@(x) (1+1/x)^(x+1)-(1+1/7)^7)
finds a function that has a vertical asymptote (to +Inf) at x=0, and a horizontal asymptote on the right as zero or at least some value below (1+1/1000000000000)^(1000000000001)-(1+1/7)^7 = 0.172023798999244, which it never reaches. Between -1 and 1 there are no real solutions.
To the left of -1, there's a horizontal asymptote just larger than 0 (larger than 1-1/100000000000000)^(-99999999999999)-(1+1/7)^7 = 0.16961033704631. It may be the same asymptote as to the right.
In any case the zero of this function is at x=-8, the only solution to this puzzle. It evaluates the given equation as (8/7)^7 = (8/7)^7.
Wofram alpha gives the horizontal asymptote as
(1/x + 1)^(x + 1) - 2097152/823543->e - 2097152/823543 as x-> ± infinity.
This is approximately 0.171782131418332, which of course is seen as e-(1+1/7)^7, Wolfrom having given a rational approximation of the second term.
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Posted by Charlie
on 2022-12-20 08:25:01 |
solution
