Multiply through by a and rearrange to get:
7a^2 + sqrt(98)a + (1/7)=0,
then it reduces to
a = (7+sqrt(47))/(7*sqrt(2)).
Given the large power in what remains, I find it hard to believe there is a strictly numerical answer, but given that we have a, and using the binomial theorem for a^x + b^x one can at least do the following:
Rewrite to a^777 + (1/a)^777, which (if I did it correctly) is:
(a + 1/a) *[ Sum k=388 to 388 of (1)^k * a^(2k)]
the series, (according to Wolfram Alpha) = (1/a)^2k
so given the 2 values for a (above), the solution seems to be
(a + 1/a) * (1/a)^776 = (1/a)^777 + (1/a)^775
not sure about simplifying further....? There could be mistakes...

Posted by Kenny M
on 20230421 12:54:13 