(Vunesp 95) A seqüência de números reais a, b, c, d forma, nessa ordem, uma progressão aritmética cuja soma dos termos é 110; a seqüência de números

A + b + c + d = 110

PA ( a, a+r, a+2r, a+3r ) -> a + (a + r ) + ( a + 2r ) + ( a + 3r ) = 110 
4a + 6r = 110

PG( a, b, e, f ) -> ( a, 2a, 4a, 8a )
b = 2a -> a + r = 2a -> r = a

PA 4a + 6a = 110 -> a = 11
P.G ( 11, 22, 44, 88 )

d + f -> 44 + 88 = 132.

\[\eqalign{ & {\text{a }} + {\text{ }}b{\text{ }} + {\text{ }}c{\text{ }} + {\text{ }}d{\text{ }} = {\text{ }}110 \cr & PA{\text{ }}\left( {{\text{ }}a,{\text{ }}a + r,{\text{ }}a + 2r,{\text{ }}a + 3r{\text{ }}} \right) \cr & a{\text{ }} + {\text{ }}\left( {a{\text{ }} + {\text{ }}r{\text{ }}} \right){\text{ }} + {\text{ }}\left( {{\text{ }}a{\text{ }} + {\text{ }}2r{\text{ }}} \right){\text{ }} + {\text{ }}\left( {{\text{ }}a{\text{ }} + {\text{ }}3r{\text{ }}} \right){\text{ }} = {\text{ }}110 \cr & 4a{\text{ }} + {\text{ }}6r{\text{ }} = {\text{ }}110 }\]

Dessa forma, temos que:

\[\eqalign{ & PG\left( {{\text{ }}a,{\text{ }}b,{\text{ }}e,{\text{ }}f{\text{ }}} \right) \cr & \left( {{\text{ }}a,{\text{ }}2a,{\text{ }}4a,{\text{ }}8a{\text{ }}} \right) \cr & b{\text{ }} = {\text{ }}2a{\text{ }} \to {\text{ }}a{\text{ }} + {\text{ }}r{\text{ }} = {\text{ }}2a{\text{ }} \to {\text{ }}r{\text{ }} = {\text{ }}a \cr & PA{\text{ }}4a{\text{ }} + {\text{ }}6a{\text{ }} = {\text{ }}110{\text{ }} \to {\text{ }}a{\text{ }} = {\text{ }}11 \cr & P.G{\text{ }}\left( {{\text{ }}11,{\text{ }}22,{\text{ }}44,{\text{ }}88{\text{ }}} \right) }\]

Então, teremos:

\[d{\text{ }} + {\text{ }}f{\text{ }} \to \boxed{44{\text{ }} + {\text{ }}88{\text{ }} = {\text{ }}132}\]
.