Sequene and series Questions Help !! (1 Viewer)

[Michaelbouts]

a) (2 points) After starting full time work a girl saves $15 in the first week, $19 in the second week, $23 in the third week and continues to increase her savings each week by the same amount until the twelfth week. How much is she then saving each week?

(b) (2 points) Find the smallest term of the sequence 1/2, 3/2, 9/2...which is greater than 300

(c) (2 points) In a small town, a census is taken at the beginning of each year. The census showed that there were 5,000 people living in the town at the beginning of 2001 and that the population decreased by 2% each year for the next seven years. How many people were living in the town 7 years later?. Write your answer to the nearest integer.

 

[aDimitri]

a) (2 points) After starting full time work a girl saves $15 in the first week, $19 in the second week, $23 in the third week and continues to increase her savings each week by the same amount until the twelfth week. How much is she then saving each week?

(b) (2 points) Find the smallest term of the sequence 1/2, 3/2, 9/2...which is greater than 300

(c) (2 points) In a small town, a census is taken at the beginning of each year. The census showed that there were 5,000 people living in the town at the beginning of 2001 and that the population decreased by 2% each year for the next seven years. How many people were living in the town 7 years later?. Write your answer to the nearest integer.

PART A
T₁ = 15
T₂ = 19
T₃ = 23
This is an Arithmetic Progression with first term 15, and common difference of 4.
Hence, according to the formula T_n = a + (n-1)d
the general term is T_n = 15 + (n-1)4
T₁₂ = 15 + 44 = $59

PART B
T₁ = 1/2
T₂ = 3/2
T₃ = 9/2
This is a Geometric Progression with first term 1/2, and common ratio of 3.
Hence, according to the formula T_n = ar^n-1
the general term is T_n = 1/2 * 3^n-1
We want to find n when T_n exceeds 300
300 < 1/2 * 3^n-1
n-1 > log₃(600)
n > ln(600)/ln(3) + 1
n > 6.8227....
n = 7
T₇ = 729/2

PART C
This is a Geometric Progression where our initial term is 5000, and each year it decreases by 2%. This is the same as multiplying it by 0.98 after each sequential year. Our general term T_n (where n is the number of years) is therefore given by:
T_n = 5000*0.98^n-1
T₇ = 5000*0.98⁶
T₇ = 4429 (to the nearest integer)