IB Mathematics SL 1.2 Arithmetic sequences and series AI SL Paper 1 - Exam Style Questions- New Syllabus
Question
In the first month of a reforestation program, the town of Neerim plants \( 85 \) trees. Each subsequent month the number of trees planted will increase by an additional \( 30 \) trees.
The number of trees to be planted in each of the first three months are shown in the following table (not to scale).
| Month | Trees planted |
|---|---|
| \( 1 \) | \( 85 \) |
| \( 2 \) | \( 115 \) |
| \( 3 \) | \( 145 \) |
(a) Obtain the number of trees to be planted in the \( 15^\text{th} \) month. [3]
(b) Obtain the total number of trees to be planted in the first \( 15 \) months. [2]
(c) Obtain the mean number of trees planted per month during the first \( 15 \) months. [2]
▶️ Answer/Explanation
Markscheme
(a)
Use the \( n^{\text{th}} \) term formula for an arithmetic sequence, \( u_n = u_1 + (n-1)d \), where \( u_1 = 85 \), \( d = 30 \), \( n = 15 \). \[ \begin{aligned} u_{15} &= 85 + (15-1) \times 30 \quad \text{(M1)} \\ &= 85 + 14 \times 30 \quad \text{(A1)} \\ &= 85 + 420 = 505 \end{aligned} \] Answer: \( 505 \) A1
[3 marks]
(b)
Use the sum formula for an arithmetic sequence, \( S_n = \frac{n}{2}(u_1 + u_n) \), where \( n = 15 \), \( u_1 = 85 \), \( u_{15} = 505 \). \[ \begin{aligned} S_{15} &= \frac{15}{2}(85 + 505) \quad \text{(M1)} \\ &= \frac{15}{2} \times 590 = 15 \times 295 = 4425 \end{aligned} \] Answer: \( 4425 \) A1
[2 marks]
(c)
The mean is the total number of trees divided by the number of months. \[ \begin{aligned} \text{Mean} &= \frac{S_{15}}{15} = \frac{4425}{15} \quad \text{(M1)} \\ &= 295 \end{aligned} \] Answer: \( 295 \) A1
[2 marks]
Total Marks: 7