2013 AMC 12A Problema 7

Question

La sucesión

S_1,

S_2,

S_3,

\ldots,

S_{10}

tiene la propiedad de que cada término a partir del tercero es la suma de los dos anteriores. Es decir,

S_n = S_{n-2} + S_{n-1} \text{ para } n \ge 3.

Supongamos que

S_9 = 110

y

S_7 = 42.

¿Cuánto vale

S_4

?

The sequence

S_1,

S_2,

S_3,

\ldots,

S_{10}

has the property that every term beginning with the third is the sum of the previous two. That is,

S_n = S_{n-2} + S_{n-1} \text{ for } n \ge 3.

Suppose that

S_9 = 110

and

S_7 = 42.

What is

S_4?

4

6

10

12

16

Reescribe la regla como

S_{n-2} = S_n - S_{n-1}

y desciende paso a paso

Rearrange the rule to

S_{n-2} = S_n - S_{n-1}

and step downward

A partir de

S_9 = S_7 + S_8

halla

S_8,

luego desciende hasta

S_4

.

From

S_9 = S_7 + S_8

find

S_8,

then work down to

S_4

Como

S_9 = S_7 + S_8,

obtenemos

S_8 = 110 - 42 = 68.

Luego

S_6 = S_8 - S_7 = 68 - 42 = 26,

S_5 = S_7 - S_6 = 42 - 26 = 16,

y

S_4 = S_6 - S_5 = 26 - 16 = 10.

Por lo tanto, la respuesta correcta es C.

Since

S_9 = S_7 + S_8,

we get

S_8 = 110 - 42 = 68.

Then

S_6 = S_8 - S_7 = 68 - 42 = 26,

S_5 = S_7 - S_6 = 42 - 26 = 16,

and

S_4 = S_6 - S_5 = 26 - 16 = 10.

Thus, the correct answer is C.

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