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
29
Number sequences
Textbook: pp. 150–152
GoalKnow what a number sequence, its term and index are; compute terms of sequences given by an n-th term formula or recurrently.
New words
number sequence · sonli ketma-ketlikterm of a sequence · ketma-ketlikning hadiformula for the n-th term · n-had formulasirecurrent (recursive) method · rekurrent usul
Explanation
If to each natural number n we assign one number a_n, we get a number sequence a₁, a₂, a₃, ..., a_n, ...: a₁ is the first term, a_n the n-th term, and n is its index. A sequence is often given by a formula for the n-th term, for example a_n = n² − 3n. To find a₁₀ we put 10 in place of n. To decide whether a number is a term, we set a_n equal to it; it is a term only if n comes out a natural number. A sequence can also be given recurrently: the first term(s) and a rule for finding each term from the previous one(s), for example a₁ = 2, a_(n+1) = 3a_n − 1. Then the terms are computed one after another.
Worked examples
a_n = n² − 3n: a₁ = −2, a₂ = −2, a₃ = 0, a₁₀ = 100 − 30 = 70. Is 28 a term? n² − 3n = 28, n² − 3n − 28 = 0, n = 7 (n = −4 is not allowed): yes, a₇ = 28.
“Chain of terms”: students stand in a line; the first says a₁, the next applies the recurrent rule and says a₂, and so on. A student who makes a mistake restarts the chain.
Practice
1
Write the first three terms of a_n = 3n − 2.
1, 4, 7
2
Find the 100th term of a_n = 3n − 2.
298
3
Is 46 a term of a_n = 3n − 2? With which index?
Yes: 3n − 2 = 46, so n = 16.
4
Why must n be a natural number when we ask whether a number is a term?
Because the indices of terms are the natural numbers 1, 2, 3, ...; there is no term with a fractional or negative index.