Because the inputs of a sequence are positions, its graph is a set of discrete points (n, an), never a connected curve. The domain of a sequence is the set of positions used: {1, 2, 3, …, N} if finite with N terms, or all whole numbers if infinite. Its range is the list of term values. The formula behind it is different: the function a(x) = a1 + d(x − 1) accepts any real x, so its domain is all real numbers, and its range is all real numbers for a linear function (d ≠ 0) or y > 0 (or y < 0 if g1 < 0) for an exponential function. So a sequence and its generating function may have different domains.
A sequence is a list of numbers in a specific order, usually following a pattern. Each number is a term. We write an for the term in position n, so a1 is the first term, a2 the second, and so on. A sequence can be infinite, shown with an ellipsis as in {1, 3, 5, 7, …}, or finite with a clear last term, as in {2, 4, 8, 16, 32}. A rule can be written two ways: an explicit rule such as an = 2n² − 3 gives any term directly from its position, while a recursive rule gives each term from the ones before it, such as a1 = 3 and an+1 = 2an − 1. Because it assigns exactly one output to each position, a sequence is a function whose domain is the whole numbers (positions 1, 2, 3, …) and whose range is a set of real numbers.
Worked Example: Find the first four terms of an = 2n² − 3, and of the recursive sequence a1 = 3, an+1 = 2an − 1.
Explicit: replace n with 1, 2, 3, 4: 2 − 3 = −1, 8 − 3 = 5, 18 − 3 = 15, 32 − 3 = 29.
Recursive: start at 3, then 2(3) − 1 = 5, 2(5) − 1 = 9, and 2(9) − 1 = 17.
Explicit: −1, 5, 15, 29. Recursive: 3, 5, 9, 17.
1. If an = n² − 2n, what is a5?
Explanation: Replace n with 5: 5² − 2(5) = 25 − 10 = 15.
2. A sequence has a1 = 2 and an+1 = 3an − 2. What is a4?
Explanation: Build term by term: a2 = 3(2) − 2 = 4, a3 = 3(4) − 2 = 10, a4 = 3(10) − 2 = 28.
3. What is the domain of a sequence?
Explanation: The input n is a position in the list, so it must be a whole number. The terms themselves are the outputs, which form the range.
4. Which sequence is finite?
Explanation: A finite sequence has a definite last term. The ellipsis in the other three signals that the pattern continues forever.
In an arithmetic sequence each term is found by adding the same number, the common difference d, to the previous term, so an − an−1 = d for every n. If a1 is the first term, then a2 = a1 + d, a3 = a1 + 2d, and in general an = a1 + d(n − 1). Notice that you add d only n − 1 times, because the first term is already in place. If the list starts at n = 0 with a0, the formula becomes an = a0 + dn. To find which term equals a given value, set the formula equal to that value and solve for n; the answer must be a whole number or the value is not a term.
Worked Example: For the arithmetic sequence 3, 7, 11, 15, … find a20, and find which term equals 55.
The first term is 3 and d = 7 − 3 = 4, so an = 3 + 4(n − 1).
a20 = 3 + 4(19) = 3 + 76 = 79.
Set 3 + 4(n − 1) = 55. Then 4(n − 1) = 52, n − 1 = 13, and n = 14.
a20 = 79; 55 is the 14th term
1. What is the 10th term of the arithmetic sequence 12, 9, 6, 3, …?
Explanation: d = 9 − 12 = −3, so an = 12 − 3(n − 1). Then a10 = 12 − 3(9) = 12 − 27 = −15.
2. An arithmetic sequence has a1 = −4 and d = 2.5. What is a9?
Explanation: a9 = −4 + 2.5(9 − 1) = −4 + 20 = 16. Using n instead of n − 1 would give the incorrect 18.5.
3. Which term of the sequence 5, 9, 13, … equals 85?
Explanation: an = 5 + 4(n − 1) = 85 gives 4(n − 1) = 80, so n = 21.
4. An arithmetic sequence has a0 = 6 and d = −2. What is a7?
Explanation: When indexing starts at 0, an = a0 + dn = 6 − 2n. So a7 = 6 − 14 = −8.
Often the problem gives two terms from the middle of a sequence instead of the first one. The general formula still works if you start from any known term ak: an = ak + d(n − k). The plan is: (1) if d is not given, use two known terms to find it with d = (an − ak)/(n − k), the change in value divided by the change in position; (2) write the rule from ak and d; (3) simplify, or evaluate at the position you want. This is the same idea as the point-slope form of a line, with (k, ak) as the point and d as the slope.
Worked Example: An arithmetic sequence has a7 = 20 and a12 = −5. Find d, a1, and a rule for an.
The terms are 5 positions apart: d = (−5 − 20)/(12 − 7) = −25/5 = −5.
Using a7: an = 20 − 5(n − 7). Setting n = 1 gives a1 = 20 − 5(−6) = 50.
Simplify: an = 20 − 5n + 35 = 55 − 5n. Check: a12 = 55 − 60 = −5. ✓
d = −5, a1 = 50, an = 55 − 5n
1. An arithmetic sequence has a5 = 17 and a9 = 33. What is a20?
Explanation: d = (33 − 17)/(9 − 5) = 4. Then a20 = 17 + 4(20 − 5) = 17 + 60 = 77.
2. An arithmetic sequence has d = −1.5 and a10 = 4. What is a30?
Explanation: a30 = 4 + (−1.5)(30 − 10) = 4 − 30 = −26.
3. An arithmetic sequence has a4 = −2 and a10 = 16. What is a1?
Explanation: d = (16 − (−2))/(10 − 4) = 3. Going back 3 positions from a4: a1 = −2 − 3(3) = −11.
4. An arithmetic sequence has a3 = 10 and a8 = −5. Which is a rule for an?
Explanation: d = (−5 − 10)/(8 − 3) = −3. Then an = 10 − 3(n − 3) = 19 − 3n. Check: a8 = 19 − 24 = −5. ✓
Sometimes two terms of an arithmetic sequence are given and you must insert a number of terms between them. If m terms are inserted between two values, then the whole list has m + 2 terms, and the jump from the first value to the last value takes (m + 2) − 1 = m + 1 steps. So d = (last − first)/(m + 1). Then add d repeatedly. A quick way to see it is to draw the list with blanks, such as 7, __, __, __, __, 32, and count the steps between the known ends.
Worked Example: Insert four terms between 7 and 32 so that all six numbers form an arithmetic sequence.
The list is 7, __, __, __, __, 32. Going from the first to the sixth term takes 5 steps, so d = (32 − 7)/5 = 5.
Add 5 each time: 7, 12, 17, 22, 27, 32.
The inserted terms are 12, 17, 22, 27
1. Insert three terms between 4 and 20 to form an arithmetic sequence. What are they?
Explanation: Four steps cover the gap of 16, so d = 16/4 = 4. The sequence is 4, 8, 12, 16, 20.
2. Three terms are inserted between −5 and 19 to form an arithmetic sequence. What is the second inserted term?
Explanation: d = (19 − (−5))/4 = 6. The sequence is −5, 1, 7, 13, 19, so the second inserted term is 7.
3. Two terms are inserted between 10 and −8 to form an arithmetic sequence. What are they?
Explanation: Three steps cover a change of −18, so d = −6. The sequence is 10, 4, −2, −8.
4. Four terms are inserted between 2 and 17 to form an arithmetic sequence. What is the third inserted term?
Explanation: Five steps cover 15, so d = 3: 2, 5, 8, 11, 14, 17. The third inserted term is 11.
In a geometric sequence each term is found by multiplying the previous term by the same nonzero number, the common ratio r. You find r by dividing any term by the one before it: r = gn/gn−1. The nth term is gn = g1·rn−1, or from any known term gn = gk·rn−k. If indexing starts at 0, gn = g0·rn. When two terms are given, divide to get r(n−k) and take the matching root. If the number of steps is even, there are two possible ratios, positive and negative. A negative ratio makes the terms alternate in sign.
Worked Example: (a) Write a rule for 80, 40, 20, 10, … and find the 8th term. (b) A geometric sequence has g1 = 5 and g5 = 80. What could g6 be?
(a) r = 40/80 = 1/2, so gn = 80(1/2)n−1. Then g8 = 80(1/2)7 = 80/128 = 0.625.
(b) 80 = 5r4, so r4 = 16 and r = 2 or r = −2. Then g6 = 80r, which is 160 or −160.
(a) g8 = 0.625. (b) g6 = 160 or −160
1. What is the 6th term of the geometric sequence 2, −6, 18, −54, …?
Explanation: r = −6/2 = −3, so g6 = 2(−3)5 = 2(−243) = −486. An odd power of a negative number is negative.
2. A geometric sequence has g1 = 81 and r = 1/3. What is g5?
Explanation: g5 = 81(1/3)4 = 81/81 = 1.
3. A geometric sequence has g3 = 12 and g6 = 96. What is g1?
Explanation: Dividing: 96/12 = r3 = 8, so r = 2. Then g1 = g3/r² = 12/4 = 3.
4. A geometric sequence has g1 = 3 and g4 = 24. What is g7?
Explanation: 24 = 3r3 gives r3 = 8 and r = 2. Then g7 = 3(2)6 = 192, which is also 24 × 8.
Treat each term as an ordered pair (n, an) and compute the average rate of change (change in term)/(change in n). For an arithmetic sequence the average rate of change between consecutive terms is always the same: it equals d. For a geometric sequence it is not constant. If the ratio is greater than 1 and the terms are positive, the rate of change keeps getting bigger, so the terms increase by larger and larger amounts. If 0 < r < 1, the terms decrease by smaller and smaller amounts and level off near 0. In short: increasing arithmetic sequences grow by equal amounts, and increasing geometric sequences grow by growing amounts.
Worked Example: Compare the arithmetic sequence 2, 5, 8, 11, … with the geometric sequence 2, 6, 18, 54, … over their first three terms.
Arithmetic: d = 3. The points are (1, 2), (2, 5), (3, 8). The rates are (5 − 2)/1 = 3 and (8 − 5)/1 = 3. Constant.
Geometric: r = 3. The points are (1, 2), (2, 6), (3, 18). The rates are (6 − 2)/1 = 4 and (18 − 6)/1 = 12. Increasing.
Arithmetic rate stays 3; geometric rate goes 4, then 12
1. What is the average rate of change between the 3rd and 4th terms of 7, 11, 15, 19, …?
Explanation: The points are (3, 15) and (4, 19), so the rate is (19 − 15)/(4 − 3) = 4. For any arithmetic sequence the rate equals d.
2. What is the average rate of change between the 3rd and 5th terms of 3, 6, 12, 24, 48?
Explanation: The points are (3, 12) and (5, 48), so the rate is (48 − 12)/(5 − 3) = 36/2 = 18. Remember to divide by the change in n.
3. Which statement is true of a geometric sequence with g1 > 0 and r > 1?
Explanation: With r > 1 each step multiplies by more than 1, and the amount added at each step is gn(r − 1), which grows with gn.
4. What is the average rate of change between the 1st and 2nd terms of 100, 50, 25, 12.5, …?
Explanation: The points are (1, 100) and (2, 50), so the rate is (50 − 100)/1 = −50. The next rate is −25, so the sequence decreases by smaller amounts each time.
Arithmetic sequences are the discrete version of linear functions, and geometric sequences are the discrete version of exponential functions.
Worked Example: Describe a sequence that fits each equation.
1. f(n) = 5 + 2n: linear form, so an arithmetic sequence with a0 = 5 and d = 2 (5, 7, 9, …).
2. g(n) = 3(1/2)n: exponential form, so a geometric sequence with g0 = 3 and r = 1/2 (3, 1.5, 0.75, …).
3. h(n) = −2(4)n−1: geometric with g1 = −2 and r = 4 (−2, −8, −32, …).
4. j(n) = 8 − 3(n − 2): arithmetic with the 2nd term equal to 8 and d = −3 (11, 8, 5, … starting at n = 1).
1. Which describes f(n) = 7 − 4n?
Explanation: 7 − 4n has the linear form b + mn, so it is arithmetic. The constant 7 is the initial term, and the coefficient −4 is the common difference.
2. Which describes g(n) = 5(2)n−1?
Explanation: The exponent n − 1 shows the form g1rn−1, so g1 = 5 (at n = 1 the factor is 20 = 1) and r = 2.
3. Which sequence matches y = 12(0.75)x?
Explanation: An exponential function a·bx corresponds to a geometric sequence with initial value a = 12 and ratio r = b = 0.75.
4. Which statement correctly contrasts linear and exponential functions?
Explanation: Equal steps in x add the same amount to a linear function but multiply an exponential function by the same factor.
Because the inputs of a sequence are positions, its graph is a set of discrete points (n, an), never a connected curve. The domain of a sequence is the set of positions used: {1, 2, 3, …, N} if finite with N terms, or all whole numbers if infinite. Its range is the list of term values. The formula behind it is different: the function a(x) = a1 + d(x − 1) accepts any real x, so its domain is all real numbers, and its range is all real numbers for a linear function (d ≠ 0) or y > 0 (or y < 0 if g1 < 0) for an exponential function. So a sequence and its generating function may have different domains.
Worked Example: Consider the sequence 3, 6, 12, 24, 48, which fits the rule gn = 3·2n−1. Give the domain and range of the sequence and of the rule viewed as a function.
The sequence has 5 terms. Domain: {1, 2, 3, 4, 5}. Range: {3, 6, 12, 24, 48}.
As a function, f(x) = 3·2x−1 allows any real x. Domain: all real numbers. Range: (0, ∞), because it is exponential with a positive initial value.
Sequence domain {1,2,3,4,5}; function domain all reals, range y > 0
1. A finite sequence has 5 terms. What is its domain?
Explanation: The domain is the set of positions used, one for each term. It is a set of whole numbers, not an interval.
2. What is the range of the function g(x) = 4(1/2)x−1?
Explanation: An exponential function with a positive initial value has outputs that are always positive and approach, but never reach, the horizontal asymptote y = 0.
3. What does the graph of a sequence look like?
Explanation: Only whole-number inputs are allowed, so the graph is a collection of discrete points, even though the points may lie along a line or curve.
4. The sequence an = 6 − 2n has 4 terms. What is its range?
Explanation: Evaluate at n = 1, 2, 3, 4: 4, 2, 0, −2. The linear function 6 − 2x would have range all real numbers, but the sequence only has these 4 outputs.
An arithmetic sequence adds the same number each step (a constant difference), and a geometric sequence multiplies by the same number each step (a constant ratio). Arithmetic sequences behave like linear functions and geometric sequences like exponential functions.
A recursive rule gives each term in terms of earlier terms, such as an+1 = an + 4, along with a starting value. An explicit rule, like an = 3 + 4(n − 1), lets you jump straight to any term.
Only a constant sequence like 5, 5, 5, … can be both: its difference is 0 and its ratio is 1. Any non-constant sequence is at most one of the two.
The Exponential and Logarithmic Functions quiz in the AP Precalculus QBank gives you more practice on this topic, and you can browse every other topic in the full AP Precalculus QBank.
