Lessons 29–30 · 2 lessons · Sh.A. Alimov, A.R. Xalmukhamedov, M.A. Mirzaakhmedov. Algebra: textbook for Grade 9 of general secondary schools. Revised 4th edition. Tashkent: “O‘qituvchi”, 2019
30
Arithmetic progression
Textbook: pp. 153–157
GoalKnow the definition of an arithmetic progression, its common difference and the n-th term formula; apply them to problems.
New words
arithmetic progression · arifmetik progressiyacommon difference · progressiya ayirmasin-th term of a progression · progressiyaning n-hadiarithmetic mean · o‘rta arifmetik
Explanation
A sequence in which each term, from the second on, is obtained from the previous one by adding the same number d is called an arithmetic progression: a_(n+1) = a_n + d, and d is the common difference (d = a_(n+1) − a_n). For example, in 7, 11, 15, ... d = 4; if d is negative the progression decreases. To prove a sequence is an arithmetic progression we show that a_(n+1) − a_n does not depend on n. The n-th term formula is a_n = a₁ + (n − 1)d. From the second term on, each term equals the arithmetic mean of its two neighbours: a_n = (a_(n−1) + a_(n+1)) : 2; this is where the name comes from. If two terms are given, we find d and a₁ from a system of two equations.
Worked examples
7, 11, 15, ...: d = 4, a_n = 7 + 4(n − 1) = 4n + 3, a₂₀ = 7 + 19 · 4 = 83. Is 99 a term? 4n + 3 = 99, n = 24, yes. Is 100 a term? 4n = 97, n is not natural, no.
“Progression hunt”: cards show sequences; groups decide which are arithmetic progressions (is the difference of neighbours always the same?) and state d.
Practice
1
State the first term and the common difference of 20, 17, 14, ...
a₁ = 20, d = −3
2
If a₁ = −4 and d = 3, find a₁₅.
38
3
Which term of 4, 7, 10, ... is 52?
17
4
Prove that a_n = 5n − 2 is an arithmetic progression.
a_(n+1) − a_n = 5(n + 1) − 2 − (5n − 2) = 5, which does not depend on n.