describe arithmetic and geometric sequences as functions, write a rule for each, and explain what the domain of a sequence is.
Instruction
A sequence is an ordered list of numbers: 7, 11, 15, 19, … Each number is a term, and each term has a position: first, second, third. That pairing of position and value is exactly what a function does. If we call the sequence a, then a(1) = 7, a(2) = 11, a(3) = 15. The input is the position n, and the output is the term. So a sequence is a function whose domain is the positive integers 1, 2, 3, … (sometimes starting at 0). You cannot ask for a(2.5); there is no term in position two and a half. That is why the graph of a sequence is a set of separate dots, not a connected line.
Arithmetic sequences. In 7, 11, 15, 19, … each term is 4 more than the one before. A sequence that adds the same number each time is arithmetic, and the number added is the common difference d. To reach position n from position 1 you add d exactly n − 1 times, so
a(n) = 7 + 4(n − 1).
Check: a(1) = 7 + 0 = 7, and a(20) = 7 + 4 × 19 = 83. Read the parts of the expression: 7 is the starting value, 4 is the step, and (n − 1) counts steps. Plotted, the dots lie on a line with slope 4, which is why arithmetic sequences are the integer version of linear functions.
Geometric sequences. In 3, 6, 12, 24, … each term is multiplied by 2. A sequence with a constant multiplier is geometric, and the multiplier is the common ratio r. To reach position n you multiply by r exactly n − 1 times:
g(n) = 3 · 2n − 1, so g(10) = 3 · 29 = 1,536.
Geometric sequences are the integer version of exponential functions. A ratio between 0 and 1 gives decay: a ball dropped from 200 cm that rebounds to 60% of its previous height reaches 120, 72, 43.2 and 25.92 cm on its first four bounces, so h(n) = 200(0.6)n, where n is the bounce number.
How to tell them apart. Subtract consecutive terms: if the differences are constant, the sequence is arithmetic. Divide consecutive terms: if the ratios are constant, it is geometric. Many sequences are neither: 1, 4, 9, 16, … has differences 3, 5, 7 and ratios 4, 2.25, 1.78. Looking for this kind of structure (MP.7) and repeated reasoning (MP.8) is the habit this unit builds.
A context. A theater’s first row has 20 seats and each row has 2 more than the row in front. Row n has s(n) = 20 + 2(n − 1) seats, so row 15 has 48. The domain is the row numbers that exist, not all real numbers, and stating that domain is part of writing the model.
Formative check
Work through these before moving on. They are not graded — they tell you, and your teacher, whether the standard below has landed yet.
The sequence 5, 10, 20, 40 continues the same way. Which is closer to the 12th term?
Which explicit formula describes 12, 9, 6, 3, …?
The graph of a sequence should be drawn as a connected curve because the rule works for any real number.
A ball is dropped from 200 cm and rebounds to 60% of its previous height each time. Write an explicit formula for the rebound height after bounce n, state its domain, and explain what each number in your formula means. On which bounce does the rebound first fall below 5 cm?
How confident are you that you can describe a sequence as a function and write an explicit rule for arithmetic and geometric sequences?
Practice
Work these on paper or in your notebook, then open Check your answer. Aim for all of Fluency and Application; try at least one Challenge.
Printable version: this unit’s practice workbook (PDF)
Fluency
Build speed and accuracy with the core skill.
- An arithmetic sequence has first term 19 and common difference 5. Write an explicit formula, then find term 10 and the sum of the first 10 terms.
Check your answer
Answer: an = 19 + (5)(n − 1); a10 = 64; S10 = 415 - An arithmetic sequence has first term 5 and common difference 6. Write an explicit formula, then find term 15 and the sum of the first 15 terms.
Check your answer
Answer: an = 5 + (6)(n − 1); a15 = 89; S15 = 705 - An arithmetic sequence has first term −6 and common difference 1. Write an explicit formula, then find term 20 and the sum of the first 20 terms.
Check your answer
Answer: an = −6 + (1)(n − 1); a20 = 13; S20 = 70 - An arithmetic sequence has first term −10 and common difference −5. Write an explicit formula, then find term 25 and the sum of the first 25 terms.
Check your answer
Answer: an = −10 + (−5)(n − 1); a25 = −130; S25 = −1750 - A geometric sequence has first term 5 and common ratio 2. Write an explicit formula, then find term 8 and the sum of the first 8 terms.
Check your answer
Answer: an = 5(2)n − 1; a8 = 640; S8 = 1275 - A geometric sequence has first term 3 and common ratio 1/2. Write an explicit formula, then find term 5 and the sum of the first 5 terms.
Check your answer
Answer: an = 3(1/2)n − 1; a5 = 3/16; S5 = 93/16 - A geometric sequence has first term 6 and common ratio 3. Write an explicit formula, then find term 9 and the sum of the first 9 terms.
Check your answer
Answer: an = 6(3)n − 1; a9 = 39366; S9 = 59046 - A geometric sequence has first term 4 and common ratio 2. Write an explicit formula, then find term 6 and the sum of the first 6 terms.
Check your answer
Answer: an = 4(2)n − 1; a6 = 128; S6 = 252 - Is 4, 9, 16, 25, … arithmetic, geometric or neither? Give the evidence.
Check your answer
Answer: Neither: the differences are 5, 7, 9 and the ratios are 2.25, about 1.78, 1.5.Check constant differences (arithmetic) and constant ratios (geometric). Neither is constant. - Is 81, 54, 36, 24, … arithmetic, geometric or neither? Write an explicit rule if you can.
Check your answer
Answer: Geometric with ratio 2/3: g(n) = 81(2/3)n − 1.54/81 = 36/54 = 24/36 = 2/3.
Application
Use the skill in context. Show your reasoning.
- A theater’s first row has 24 seats and each row has 3 more than the row in front. How many seats are in row 18, and which is the first row with at least 100 seats?
Check your answer
Answer: Row 18 has 75 seats; row 27 is the first with at least 100.s(n) = 24 + 3(n − 1). s(18) = 24 + 51 = 75. Solve 24 + 3(n − 1) ≥ 100: n ≥ 26.33, so row 27. - A ball is dropped from 150 cm and each rebound reaches 70% of the previous height. Find the height of the 5th rebound and the first rebound lower than 10 cm.
Check your answer
Answer: About 25.21 cm; rebound 8 is the first below 10 cm.h(n) = 150(0.7)n. h(5) = 150(0.16807) ≈ 25.21. h(7) ≈ 12.35 and h(8) ≈ 8.65. - A Fresno nursery (illustrative figures) plants 40 almond seedlings in week 1 and 15 more each week than the week before. How many does it plant in week 20? Is the domain all real numbers?
Check your answer
Answer: 325 seedlings in week 20. No: the domain is the week numbers 1, 2, 3, …a(n) = 40 + 15(n − 1), so a(20) = 40 + 285 = 325.
Challenge
Stretch problems. Expect to think before you write.
- The sequence 3, 7, 13, 21, … is neither arithmetic nor geometric. Its differences are 4, 6, 8. Find an explicit rule and the 10th term.
Check your answer
Answer: a(n) = n2 + n + 1; a(10) = 111.Constant second differences suggest a quadratic. n2 + n + 1 gives 3, 7, 13, 21 for n = 1 to 4.
Review
Keep earlier skills sharp.
- If f(x) = 3x2 − 2x + 1, find f(−3).
Check your answer
Answer: f(−3) = 34Substitute, then follow the order of operations; square the input first. - If f(x) = 2x2 − 4x − 5, find f(3).
Check your answer
Answer: f(3) = 1Substitute, then follow the order of operations; square the input first. - Simplify: (x2)5
Check your answer
Answer: x10Power of a power: multiply exponents. - Simplify: x7 · x5
Check your answer
Answer: x12Same base: add exponents.