If-u-5-v-5-u-v-5-1-5-find-u-3-v-3-u-v-3-

Question Number 54607 by ajfour last updated on 07/Feb/19

I f \frac{u^{5} + v^{5}}{{(u + v)}^{5}} = - \frac{1}{5},

f i n d \frac{u^{3} + v^{3}}{{(u + v)}^{3}} = ?

Commented by MJS last updated on 08/Feb/19

I think it can be - \frac{4}{5} or - \frac{1}{5}

Commented by ajfour last updated on 08/Feb/19

y e s s i r, i g e t - \frac{1}{5} .

Answered by mr W last updated on 08/Feb/19

l e t \frac{v}{u} = t

\frac{u^{5} + v^{5}}{{(u + v)}^{5}} = \frac{1 + t^{5}}{{(1 + t)}^{5}} = \frac{1 - t + t^{2} - t^{3} + t^{4}}{{(1 + t)}^{4}} = - \frac{1}{5}

6 - t + 11 t^{2} - t^{3} + 6 t^{4} = 0

(2 t^{2} - t + 2) (3 t^{2} + t + 3) = 0

\Rightarrow t^{2} - \frac{1}{2} t + 1 = 0 \dots (i)

\Rightarrow t^{2} + \frac{1}{3} t + 1 = 0 \dots (i i)

l e t \frac{u^{3} + v^{3}}{{(u + v)}^{3}} = \frac{1 + t^{3}}{{(1 + t)}^{3}} = \frac{1 - t + t^{2}}{{(1 + t)}^{2}} = \frac{1}{s}

s - s t + {s t}^{2} = 1 + 2 t + t^{2}

\Rightarrow t^{2} + \frac{2 + s}{1 - s} t + 1 = 0

w i t h \frac{2 + s}{1 - s} = - \frac{1}{2}

\Rightarrow s + 5 = 0 \Rightarrow s = - 5

w i t h \frac{2 + s}{1 - s} = \frac{1}{3}

\Rightarrow 4 s + 5 = 0 \Rightarrow s = - \frac{5}{4}

\Rightarrow \frac{u^{3} + v^{3}}{{(u + v)}^{3}} = \frac{1}{s} = - \frac{1}{5} o r - \frac{4}{5}

Commented by ajfour last updated on 08/Feb/19

T h a n k s S i r, y o u h a v e d o n e b e t t e r .

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