Find-the-value-of-m-given-that-the-roots-of-x-4-15x-3-70x-2-120x-m-0-form-a-geometric-progression-

Question Number 198932 by necx122 last updated on 25/Oct/23

F i n d t h e v a l u e o f m g i v e n t h a t t h e

r o o t s o f x^{4} - 15 x^{3} + 70 x^{2} - 120 x + m = 0

f o r m a g e o m e t r i c p r o g r e s s i o n .

Answered by AST last updated on 25/Oct/23

m = a b c d = ({b r}^{- 1}) b (b r) ({b r}^{2}) = b^{4} r^{2}

a + b + c + d = 15 \Rightarrow a (1 + r + r^{2} + r^{3}) = 15 \dots (i)

a b + b c + c d + d a + c a + b d = 70

\Rightarrow a^{2} (r + r^{3} + r^{5} + r^{3} + r^{2} + r^{4}) = 70 \dots (i i)

a b c + b c d + a c d + a b d = b^{3} (1 + r^{3} + r^{2} + r) = 120 \dots (i i i)

(i) & (i i i) \Rightarrow \frac{120}{b^{3}} = \frac{15}{a} = 8 a = b^{3}

b^{3} = a^{3} r^{3} = 8 a \Rightarrow a^{2} r^{3} = 8

\Rightarrow m = b^{4} r^{2} = a^{4} r^{6} = {(a^{2} r^{3})}^{2} = 64

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