Question

If `sin^(-1)x_1+sin^(-1)x_2++sin^(-1)x_nlt==(npi)/2,n in N ,n=2m+1, mgeq1,` then find the value of `(x1 1+x3 3+x5 5+(m+1)t e r m s)/(x2 2+x4 4+x6 6+ mt e r m s)`

Answer 1

Since `sin^(-1) x_(i) ge - (pi)/(2), I = 1, 2, 3, ...., n`,
`sin^(-1) x_(1) + sin^(-1) x_(2) + ... + sin^(-1) x_(n) ge - (npi)/(2)`
But given that `underset(i = 1) overset(n) sum sin^(-1) x_(i) le - (npi)/(2)`
`sin^(-1) x_(1) = sin^(-1) x_(2) = ... = sin^(-1) x_(n) = -(pi)/(2)`
`:. x_(i) = -1, i = 1, 2, 3, ..., n`
`:. (x_(1)^(1) + x_(3)^(3) + x_(5)^(5) + ...(m + 1) " terms")/(x_(2)^(2) + x_(4)^(4) + x_(6)^(6) + m... "terms")`
`= ((-1) + (-1) + (-1) + ..(m + 1) "times")/(1 + 1 + 1 + ...m "times")`
`= (-(m + 1))/(m)`