Question

Determine the values of a,b that belong to the real numbers such that the sequence a,8,b is in arithmetic progression and the sequence a,8,b+4 is in geometric progression.

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Hermann

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118 Answers

a, 8, b es una secuencia aritmética
=> 8 es la media aritmética de a y b
\Flecha derecha8\:=\:\frac{a+b}{2}
\Flecha derecha a+b=16
\Flecha derecha a=16-b\:\ldots\ldots\ldots\ldots..\:\left(i\right)
a, 8, (b+ 4) es una secuencia geométrica
=> 8 es la media geométrica de a y (b + 4)
\Flecha derecha\:8^2=a\izquierda(b+4\derecha)
\Flecha derecha a\left(b+4\right)=64
\Flecha derecha\izquierda(16-b\derecha)\izquierda(b+4\derecha)=64\:\:\:\izquierda(desde\:\izquierda(i\derecha)\derecha)
\Flecha derecha16b+64-b^2-4b=64
\Flecha derecha12b-b^2=0
\Flecha derecha b\izquierda(12-b\derecha)=0
\Flecha derecha b=0\:o\:b=12
b = 0 da a = 16 - b = 16
b = 12 da a = 16 - 12 = 4
Las soluciones son (a = 16, b = 0) y (a = 4, b = 12)

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