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Lesson 4.03
Madame Dumas has a rather extensive art collection and the overall value of her collection has been
increasing each year. Three years ago, her collection was worth $250,000. Two years ago, the value of the
collection was $525,000 and last year, the collection was valued at $1,102,500.
Assume that the rate at which Madame Dumas's art collection's value increase remains the same as it has
been for the last three years. The value of the art collection can be represented by a geochetric sequence.
The value of the collection three years ago is considered the first term in the sequence.
A) Write an explicit rule which can be used to determine the value of her art collection n years after that.
B) Use this rule to determine the value of her collection 9 years after she started tracking its worth rounded
to the nearest dollar

Lesson 403 Madame Dumas Has A Rather Extensive Art Collection And The Overall Value Of Her Collection Has Been Increasing Each Year Three Years Ago Her Collect class=

Sagot :

Using a geometric sequence, it is found that:

a) The explicit formula is: [tex]a_n = 250000(2.1)^{n-1}[/tex].

b) The value of her collection 9 years after she started tracking is of $94,557,148.

What is a geometric sequence?

A geometric sequence is a sequence in which the result of the division of consecutive terms is always the same, called common ratio q.

The nth term of a geometric sequence is given by:

[tex]a_n = a_1q^{n-1}[/tex]

In which [tex]a_1[/tex] is the first term.

Item a:

Three years ago, her collection was worth $250,000, hence the first term is [tex]a_1 = 250000[/tex].

In two years, the collection was valued at $1,102,500, hence the common ratio is:

[tex]q^2 = \frac{1,102,500}{250,000}[/tex]

[tex]q^2 = 4.41[/tex]

[tex]q = \sqrt{4.41} = 2.1[/tex].

Hence, the explicit formula is:

[tex]a_n = 250000(2.1)^{n-1}[/tex]

Item b:

[tex]a_9 = 250000(2.1)^{9-1} = 94557148[/tex]

The value of her collection 9 years after she started tracking is of $94,557,148.

More can be learned about geometric sequences at https://brainly.com/question/11847927

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