The value of an antique increases at a rate of 3.5% every year. In 2000, the antique was purchased for $5000.
a.) Determine an explicit formula to represent the value of the antique since year 2000.
b.) Use your formula to write the first three terms of the sequence.
c.) What is the value of the antique in 2008?
d.) In which year will the value of the antique be $20489?
Answers
Step-by-step explanation:
Given :-
The value of an antique increases at a rate of 3.5% every year. In 2000, the antique was purchased for $5000.
To find :-
Find the following :-
a.) Determine an explicit formula to represent the value of the antique since year 2000.
b.) Use your formula to write the first three terms of the sequence.
c.) What is the value of the antique in 2008?
d.) In which year will the value of the antique be $20489?
Solution :-
(a)
The Cost Price of an antique in the year 2000= $5000
Increasing percentage of the antique every year = 3.5%
Increasing value = 3.5% of Cost Price
=> 3.5% of 5000
= (3.5/100)×5000
= 3.5×50
= 35×5
= 175
Increasing value = 175
The value of the antique in the year 2001
=5000+175
=$ 5175
The Increasing value after 1 year
= 3.5% of 5175
=> (3.5/100)×5175
=> (35/1000)×5175
= 181.125
The cost price of the antique in 2002
= 5175 + 181.125
= $ 5356.125
We have ,
5000,5175 , 5356.125,...
First term = 5000
Common ratio
= 5175/5000 = 1.035
= 5356.125/5175 = 1.035
Since the Common ratio is same throughout the series
They are in the GP
We know that
The general form of a GP
= an = a×r^(n-1)
=>an = 5000×(1.035)^(n-1)
The Formula = 5000×(1.035)^(n-1)-------(1)
(b)
Put n = 2 then
a2 = 5000×(1.035)^(2-1)
=> a2 = 5000×1.035
=> a2 = 5175
Put n = 3 then
a3 = 5000×(1.035)^(3-1)
=>a3 = 5000×(1.035)²
=> a3 = 5000×1.035×1.035
=> a3 = 5356.125
The first three terms of the sequence are 5000, 5175,5356.125
(c)
Given year = 2008
The initial value in 2000 = $5000
Number of years = 2008-2000+1 = 9
Put n = 9 in (1) then
a9 = 5000×(1.035)^(9-1)
=> a9 = 5000×(1.035)⁸
=> a9 = 5000×1.3168090369634
=> a9 = 6584.0451848170
=> a9 = 6584.045
The value of the antique in the year 2008 is
The value of the antique in the year 2008 is $ 6584.045
(d)
Given cost of the antique = $20489
Let an = $20489
=> a×r^(n-1) = $20489
=> 5000×(1.035)^(n-1) = 20489
=> (1.035)^(n-1) = 20489/5000
=> (1.035)^(n-1) = 4.0978
=> (1.035)^(n-1) = (1.035)⁴¹
Since the bases are equal then exponents must be equal
=> n-1 = 41
=> n = 41+1
=> n = 42
2000 year + 42 years= 2041 year
In 2041 the Cost of the antique is
$ 20489
Answer:-
(a)an explicit formula to represent the value of the antique since year 2000 is 5000×(1.035)^(n-1)
(b)The first three terms of the sequence are 5000, 5175,5356.125
(c) The value of the antique in the year 2008 is $ 6584.045
(d) In 2041 the Cost of the antique is
$ 20489
Used formulae:-
- The general term of a GP is a×r^(n-1)
- a = first term
- r = Common ratio
- n = number of terms