4. a) A group of studets decided to collect old newspapers to raise fund for charity. On the first day, they collected 41 kg of old newspapers. In each subsequent day, they collected 19 kg of old newspapers more than the previous day. Calculate the total mass of old newspapers collected in the first five weeks. [ 4 marks]
b) In a nursery, a flower plant grows 2.15 cm in the first week. For each subsequent week, it grows by 5% taller than the previous week.
i) Write down the pattern for the successive growth of the flower plant in the first n weeks. [ 5 marks]
ii) How much does the flower plant grow in 15 weeks? Give your answer in cm correct to two decimal places [3 marks]
Answers
Given : On the first day, they collected 41 kg of old newspapers. In each subsequent day, they collected 19 kg of old newspapers more than the previous day
To Find : the total mass of old newspapers collected in the first five weeks.
Solution :
Arithmetic sequence : Sequence of terms in which difference between one term and the next is a constant.
This is also called Arithmetic Progression AP
Arithmetic sequence can be represented in the form :
a, a + d , a + 2d , …………………………, a + (n-1)d
a = First term
d = common difference = aₙ-aₙ₋₁
nth term = aₙ = a + (n-1)d
Sum of Arithmetic sequence (AP) is called Arithmetic series
Sₙ = (n/2)(2a + (n - 1)d) or (n/2)(a + nth term)
a = 41
d = 15
n = 5
Sum = (5/2)( 2 * 41 + (5 - 1)19)
= 5 ( 41 + 38)
= 5(79)
= 395
total mass of old newspapers collected in the first five weeks = 395 kg
Geometric sequence
A sequence of numbers in which the ratio between consecutive terms is constant and called the common ratio.
a , ar , ar² , ... , arⁿ⁻¹
The nth term of a geometric sequence with the first term a and the common ratio r is given by: aₙ = arⁿ⁻¹
Sum is given by Sₙ = a(rⁿ - 1)/(r - 1)
Sum of infinite series is given by a/(1 - r) where -1 < r < 1
Geometric Series
The sum of the terms of a geometric sequence is called a geometric series.
a = 2.15
r = 1.05
2.15 , 2.15(1.05) , (2.15)(1.05)² , ... . (2.15)(1.05)ⁿ⁻¹
flower plant grow in 15 weeks
= (2.15)(1.05)¹⁵⁻¹
= (2.15)(1.05)¹⁴
≈ 4.257 cm
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