IBDP Maths AA HL – SL 1.2 Arithmetic Sequences & Series HL Paper 1 Exam Style Questions
IB MATHEMATICS AA HL – Practice Questions- Paper1-All Topics
Topic : SL 1.2 Arithmetic Sequences & Series HL Paper 1
Topic 1- Number and algebra- Weightage : 15 %
All Questions for Topic : SL 1.2 Arithmetic Sequences & Series HL Paper 1. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for sums of arithmetic sequences. Applications. Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.. approximate common differences.
▶️ Answer/Explanation
(a)
Given: \( S_n = pn^2 – qn \).
For \( n = 4 \): \( S_4 = 16p – 4q = 40 \implies 4p – q = 10 \) (1).
For \( n = 5 \): \( S_5 = 25p – 5q = 65 \implies 5p – q = 13 \) (2).
Subtract (1) from (2): \( (5p – q) – (4p – q) = 13 – 10 \implies p = 3 \).
Substitute \( p = 3 \) in (1): \( 4(3) – q = 10 \implies 12 – q = 10 \implies q = 2 \).
Thus, \( p = 3 \), \( q = 2 \).
(b)
\( u_n = S_n – S_{n-1} \).
\( S_n = 3n^2 – 2n \), \( S_{n-1} = 3(n-1)^2 – 2(n-1) = 3n^2 – 6n + 3 – 2n + 2 = 3n^2 – 8n + 5 \).
\( u_n = (3n^2 – 2n) – (3n^2 – 8n + 5) = 6n – 5 \).
For \( n = 5 \): \( u_5 = 6(5) – 5 = 25 \).
Markscheme
(a) Using \( S_4 = 40 \), \( S_5 = 65 \):
\( 16p – 4q = 40 \), \( 25p – 5q = 65 \).
Solving: \( p = 3 \), \( q = 2 \).
(b) \( u_5 = S_5 – S_4 = 65 – 40 = 25 \).