Arithmetic sequences and the explicit formula
Each row of an amphitheater holds four more seats than the row in front. How many seats are in row 30, without counting through rows 2 to 29?
Where you are 10 in Sequences and series
When you reach for it
An arithmetic sequence is a sequence in which every term is the term before it plus the same fixed number. That fixed number is the common difference, written d:
The common difference may be positive (the terms climb), negative (they fall), zero (every term is the same) or a fraction. Terms, positions and the notation a_n are set out in Sequences, terms and notation.
Reach for the explicit formula when you want a term far down the list without building every term before it, when you want to know whether a given number appears in the sequence and in which position, or when you want to treat the sequence as a function of its position and read it the way you read any constant rate of change.
The procedure
Finding the common difference. Subtract each term from the one after it. The front row of an amphitheater has 18 seats and each row behind has 4 more:
The differences are 22 - 18 = 4, 26 - 22 = 4, and so on, so d = 4. The order of the subtraction matters: it is always the later term minus the earlier one, which is why the falling sequence 7, 4, 1, -2 has d = -3 and not 3. If the differences are not all equal, as in 1, 2, 4, 8 with differences 1, 2, 4, the sequence is not arithmetic and nothing below applies.
Building the formula. Write each term as the first term plus however many fours have been added to it.
| Row n | Seats a_n | Built from a_1 |
|---|---|---|
| 1 | 18 | 18 + 0 \cdot 4 |
| 2 | 22 | 18 + 1 \cdot 4 |
| 3 | 26 | 18 + 2 \cdot 4 |
| 4 | 30 | 18 + 3 \cdot 4 |
| n | a_n | 18 + (n - 1) \cdot 4 |
The number of fours is always one less than the row number. The first row has its seats before any four is added, and getting from row 1 to row n takes n - 1 steps, just as walking from the first fence post to the nth passes n - 1 gaps. The same count works for any arithmetic sequence and gives the explicit formula
Row 30 is 29 steps behind row 1, so a_{30} = 18 + 29 \cdot 4 = 134. No row in between had to be worked out.
The same rule as a line. Multiply out the bracket and collect the constants:
This is a linear function of n in slope-intercept form, with slope d and intercept a_1 - d. For the seats it is a_n = 4n + 14. The slope is the common difference for a plain reason: from one term to the next, n grows by 1 and a_n grows by d, and a rise of d for a run of 1 is a slope of d. The intercept 14 is what the rule gives at n = 0, one step before the first term. There is no row 0, so 14 is not a term; it is only where the line through the dots meets the vertical axis.
The graph of a sequence is the dots alone, one for each position n = 1, 2, 3, \dots. The dashed line is drawn only to show that they all sit on one line.
Running it backwards. To find which row has 70 seats, set the rule equal to 70 and solve for n: 4n + 14 = 70 gives 4n = 56 and n = 14, so row 14. To ask whether any row has exactly 100 seats, 4n + 14 = 100 gives n = 21.5. A position is a positive whole number, so no row has 100 seats: row 21 has 98 and row 22 has 102. A value is a term exactly when the n it gives is a positive whole number.
From two terms that are not next to each other. Suppose a_4 = 20 and a_{10} = 56. From position 4 to position 10 there are 10 - 4 = 6 steps, and together they climb 56 - 20 = 36, so each step is d = 36 \div 6 = 6. This is the slope between the points (4, 20) and (10, 56), found as in The slope formula and the four kinds of slope. Then walk back three steps from a_4 to the first term: a_1 = 20 - 3 \cdot 6 = 2. So
and the check 6 \cdot 10 - 4 = 56 agrees with the given a_{10}.
Worked examples
Deciding whether a sequence is arithmetic
Decide whether each sequence is arithmetic. If it is, give d and the next two terms.
(a) 23, 17, 11, 5, \dots \qquad (b) \displaystyle \frac{3}{4}, \frac{5}{4}, \frac{7}{4}, \frac{9}{4}, \dots \qquad (c) 2, 5, 9, 14, \dots
Solution(a) The differences are 17 - 23 = -6, 11 - 17 = -6 and 5 - 11 = -6. They are all equal, so the sequence is arithmetic with d = -6. The next two terms are 5 - 6 = -1 and -1 - 6 = -7.
(b) Each difference is \displaystyle \frac{2}{4} = \frac{1}{2}, so \displaystyle d = \frac{1}{2}. The next two terms are \displaystyle \frac{11}{4} and \displaystyle \frac{13}{4}.
(c) The differences are 3, 4, 5. They are not all equal, so the sequence is not arithmetic. It does have a pattern, since the gaps grow by one each time, but no single number is added at every step.
Chairs in a stack
Ramona stacks chairs that nest inside each other. One chair stands 82 cm tall, and each chair added to the stack raises it by 6 cm.
(a) Write the height h_n of a stack of n chairs as an explicit formula, and as a linear function of n. (b) How tall is a stack of 15 chairs? (c) What is the largest stack that fits through a doorway 200 cm high?
Solution(a) The heights 82, 88, 94, \dots form an arithmetic sequence with h_1 = 82 and d = 6, so
The slope 6 is centimeters per chair. The 76 is the value of the rule at n = 0, which is not a stack at all; it is the part of one chair’s height that nesting never repeats, since 82 = 76 + 6.
(b) h_{15} = 6 \cdot 15 + 76 = 166 cm. The explicit form agrees: 82 + 14 \cdot 6 = 82 + 84 = 166.
(c) The stack fits when 6n + 76 \le 200, that is 6n \le 124, so n \le 20.66\ldots The number of chairs is a whole number, so the largest stack is 20 chairs, 6 \cdot 20 + 76 = 196 cm tall. A 21st chair would make it 202 cm.
A countdown that falls by a fraction
A countdown display starts at 10 and drops by \displaystyle \frac{3}{4} at each tick, so it shows 10, 9\tfrac{1}{4}, 8\tfrac{1}{2}, 7\tfrac{3}{4}, \dots
(a) Write the explicit formula. (b) Which term is 1? (c) Does the display ever show exactly 0?
Solution(a) The terms fall, so the common difference is negative: a_1 = 10 and \displaystyle d = -\frac{3}{4}.
The constant is \displaystyle 10 + \frac{3}{4} = \frac{43}{4}.
(b) Set \displaystyle -\frac{3}{4}n + \frac{43}{4} = 1 and multiply both sides by 4 to clear the fractions: -3n + 43 = 4, so 3n = 39 and n = 13. The 13th term is 1. As a check, twelve drops of \displaystyle \frac{3}{4} total 9, and 10 - 9 = 1.
(c) Setting the rule equal to 0 gives -3n + 43 = 0, so \displaystyle n = \frac{43}{3} = 14\tfrac{1}{3}. That is not a whole number, so 0 is not a term. The display steps over it: \displaystyle a_{14} = \frac{1}{4} and \displaystyle a_{15} = -\frac{1}{2}.
Which term, and the first positive one
The sequence -40, -33, -26, \dots is arithmetic.
(a) Is 100 a term, and if so which one? (b) Is 150 a term? (c) Which is the first positive term?
SolutionThe common difference is -33 - (-40) = 7, so a_n = -40 + 7(n - 1) = 7n - 47.
(a) 7n - 47 = 100 gives 7n = 147 and n = 21. So 100 is the 21st term.
(b) 7n - 47 = 150 gives 7n = 197 and n = 28.14\ldots, not a whole number, so 150 is not a term. It falls between a_{28} = 149 and a_{29} = 156.
(c) A term is positive when 7n - 47 > 0, that is when \displaystyle n > \frac{47}{7} = 6\tfrac{5}{7}. The smallest whole number above that is 7, and a_7 = 49 - 47 = 2. The term before it, a_6 = -5, is still negative.
Two terms and a falling, fractional difference
An arithmetic sequence has a_3 = 11 and a_{15} = -7. Find d, a_1, the explicit formula and a_{25}.
SolutionFrom position 3 to position 15 there are 15 - 3 = 12 steps, and the terms change by -7 - 11 = -18. So
The first term is two steps back from a_3. Going back a step subtracts d, and subtracting \displaystyle -\frac{3}{2} adds \displaystyle \frac{3}{2}, so \displaystyle a_1 = 11 + 2 \cdot \frac{3}{2} = 14. Then
Check against the other given term: \displaystyle -\frac{45}{2} + \frac{31}{2} = -\frac{14}{2} = -7, which is a_{15}. Finally \displaystyle a_{25} = -\frac{75}{2} + \frac{31}{2} = -\frac{44}{2} = -22.
Three expressions in arithmetic order
For which value of x are x, 2x + 1 and 5x - 4 three consecutive terms of an arithmetic sequence? Give the three terms and the common difference.
SolutionConsecutive terms of an arithmetic sequence differ by the same amount, so the second minus the first must equal the third minus the second:
The left side is x + 1 and the right side is 3x - 5. Setting them equal gives 6 = 2x, so x = 3. The terms are 3, 7, 11 and the common difference is 4. Notice that the middle term is the mean of its neighbors, \displaystyle 7 = \frac{3 + 11}{2}: that holds for any three consecutive terms, because the middle one sits the same distance d from each.
Boundaries
Positions are positive whole numbers. The rule a_n = dn + (a_1 - d) can be evaluated at n = 0 or n = 2.5, but those give points on the line, not terms. When solving for n gives a fraction, zero or a negative number, the value is not a term.
The formula needs every difference to be d. A few listed terms with equal differences do not force the next one: 2, 4, 6 could go on to 8 or to 10. The formula describes a sequence only once it is known, or stated, to be arithmetic.
The (n - 1) counts from a_1. If a sequence is numbered from a_0, then position n is n steps from the start and the formula is a_n = a_0 + nd.
A common difference of 0 gives a constant sequence, whose dots lie on a horizontal line. Its one value is every term, so “which term is it” has no single answer, and every other value is no term at all.
Two given terms must be in different positions. Finding d divides the change in value by the number of steps between them, and that number cannot be zero.
Where this goes next
The seats can also be described by saying how each row comes from the one in front: a_1 = 18 and a_n = a_{n-1} + 4. That is a recursive formula, and Closed form versus recursive form weighs it against the explicit one. Change “add the same number” to “multiply by the same number” and the dots stop lying on a line: those are geometric sequences, which grow exponentially. Equal differences are also the signal of a linear pattern in data, which Choosing a function family for data uses. Adding up the terms, every seat in the amphitheater at once, is the subject of arithmetic series.
Exercises
The common difference
Decide whether each sequence is arithmetic. If it is, give d.
- 31, 24, 17, 10, \dots
- \displaystyle \frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \dots
- 1, 4, 9, 16, \dots
- Write the first five terms of the arithmetic sequence with a_1 = -6 and \displaystyle d = \frac{5}{2}.
Writing the formula and finding a term
- An arithmetic sequence has a_1 = 9 and d = 4. Write its explicit formula and find a_{40}.
- Write the explicit formula for 50, 46, 42, \dots and find a_{30}.
- Write the explicit formula for 2.5, 3.25, 4, \dots and find a_{21}.
- An amphitheater has 20 rows. The front row has 24 seats and each row has 3 more than the row in front. How many seats are in the back row? Which row has 60 seats? Does any row have exactly 50?
Which term
- Which term of 5, 12, 19, \dots is 236?
- For 100, 94, 88, \dots, is 0 a term? Is -2 a term? Give the position of any that are.
Two terms that are not next to each other
- An arithmetic sequence has a_5 = 26 and a_{12} = 61. Find d, a_1 and the explicit formula.
- An arithmetic sequence has a_2 = -4 and a_8 = 5. Find the explicit formula and a_{20}.
- Nicasio measures a stack of 4 nesting chairs at 101 cm and a stack of 10 at 131 cm. How tall is one chair, and what is the largest stack that fits under a 180 cm shelf?
- A countdown starts at 20 and falls by the same amount at each step, and its 25th term is 4. Find the explicit formula. Does the countdown ever show exactly 0?
The sequence as a line
- Which of these rules give arithmetic sequences? Give d for each one that does. (a) a_n = 3 - 5n \qquad (b) a_n = n^2 + 1 \qquad (c) \displaystyle a_n = \frac{n + 7}{2}
- The first three terms of an arithmetic sequence, plotted as points (n, a_n), are (1, 9), (2, 6.5) and (3, 4). Give the slope and intercept of the line they lie on, write a_n as a linear function of n, and find which term is -16.
Reasoning
- Suppose the formula were written as a_n = a_1 + nd. Use n = 1 to show that this version is wrong, and explain in a sentence or two why the correct formula has n - 1.
- Show that any rule a_n = mn + b gives an arithmetic sequence with common difference m. Then explain why the dots of a sequence whose differences are not all equal can never lie on one straight line.
Answers
- Arithmetic, d = -7. Each difference is -7.
- Arithmetic, \displaystyle d = \frac{2}{3}. Write 1 as \displaystyle \frac{3}{3}: each step adds \displaystyle \frac{2}{3}.
- Not arithmetic. The differences are 3, 5, 7.
- \displaystyle -6, -\frac{7}{2}, -1, \frac{3}{2}, 4. Each term adds \displaystyle \frac{5}{2} to the one before.
- a_n = 9 + 4(n - 1) = 4n + 5, and a_{40} = 4 \cdot 40 + 5 = 165.
- d = -4, so a_n = 50 - 4(n - 1) = -4n + 54, and a_{30} = -120 + 54 = -66.
- d = 0.75, so a_n = 2.5 + 0.75(n - 1) = 0.75n + 1.75, and a_{21} = 2.5 + 20 \cdot 0.75 = 17.5.
- a_n = 24 + 3(n - 1) = 3n + 21. The back row has a_{20} = 81 seats. 3n + 21 = 60 gives n = 13, so row 13. No row has 50: 3n + 21 = 50 gives n = 9\tfrac{2}{3}, and rows 9 and 10 have 48 and 51.
- The 34th. Here a_n = 7n - 2, and 7n - 2 = 236 gives 7n = 238, n = 34.
- a_n = -6n + 106. 0 is not a term: 6n = 106 gives n = 17\tfrac{2}{3} (the terms step from a_{17} = 4 to a_{18} = -2). -2 is a term: 6n = 108 gives n = 18.
- d = 5, a_1 = 6, a_n = 5n + 1. There are 7 steps and a change of 35, so d = 5; then a_1 = 26 - 4 \cdot 5 = 6.
- \displaystyle a_n = \frac{3}{2}n - 7 and a_{20} = 23. Six steps climb 9, so \displaystyle d = \frac{3}{2}, and \displaystyle a_1 = -4 - \frac{3}{2} = -\frac{11}{2}; then \displaystyle a_n = -\frac{11}{2} + \frac{3}{2}(n - 1).
- One chair is 86 cm, and the largest stack is 19 chairs. Six chairs add 30 cm, so each adds 5 cm and h_1 = 101 - 3 \cdot 5 = 86. Then h_n = 5n + 81, and 5n + 81 \le 180 gives n \le 19.8; 19 chairs stand 176 cm.
- \displaystyle a_n = -\frac{2}{3}n + \frac{62}{3}, and yes: a_{31} = 0. The 24 steps fall 16, so \displaystyle d = -\frac{2}{3} and \displaystyle a_n = 20 - \frac{2}{3}(n - 1). Setting it to 0 gives 2n = 62.
- (a) Arithmetic, d = -5. (b) Not arithmetic: the terms 2, 5, 10, 17 have differences 3, 5, 7. (c) Arithmetic, \displaystyle d = \frac{1}{2}: the terms are \displaystyle 4, \frac{9}{2}, 5, \dots
- Slope -2.5, intercept 11.5, so a_n = -2.5n + 11.5. Setting -2.5n + 11.5 = -16 gives 2.5n = 27.5, so -16 is the 11th term.
- With n = 1 that version gives a_1 = a_1 + d, which is false unless d = 0. The first term has had no difference added to it yet, and term n is reached after n - 1 steps from term 1, so exactly n - 1 copies of d have been added.
- a_{n+1} - a_n = m(n + 1) + b - (mn + b) = m, the same for every n, so the sequence is arithmetic with d = m. Each difference is also the slope between neighboring dots, since their positions are 1 apart. Points on one non-vertical line all give the same slope, so if two of the differences are unequal, the dots cannot share one line.