A sequence is an ordered list of numbers: a first term, a second term, and so on. In this course we treat a sequence as a function. The inputs are the whole numbers 0, 1, 2, 3, … (the position in the list) and the outputs are real numbers (the terms).
We write the output for input n as an (or gn) instead of a(n). Counting starts at n = 0, so a0 is the initial value: the starting amount in a context, like the balance on day 0.
input n (whole number) → output an (real number)
Pick a sequence, then slide the input n. The input has to be a whole number, so the slider only stops at whole numbers, and the graph is a set of separate points.
input n3
a3 = 9 (the term at position n = 3)terms: 3, 5, 7, 9, 11, 13, 15, …There is no a3.5: the input has to be a whole number, so nothing sits between the points.
⚠️Do not connect the dots. The inputs are whole numbers only, so there is no term at n = 2.5. The graph of a sequence is a set of separate points, not a curve.
💡Because counting starts at 0, a3 is the fourth term in the list. The subscript is the input n, not the number of terms so far.
2.1.A2.1.A.2Concept
Arithmetic Sequences and the Common Difference
In an arithmetic sequence, successive terms have a common difference: the same amount is added at every step. That makes the common difference d a constant rate of change. Each time the input n goes up by 1, the output changes by d.
d = an+1 − an (the same for every n)
The difference d is…
The sequence is…
Example
positive
increasing
5, 9, 13, 17, … (d = 4)
negative
decreasing
20, 17, 14, 11, … (d = −3)
Pick a list of terms. The checker subtracts each term from the next one. A sequence is arithmetic only if every one of those differences is the same.
differences (each term − the one before): 4, 4, 4, 4✓ Arithmetic: the common difference is d = 4.
🔑To show a sequence is arithmetic, check every pair of successive terms. Two equal differences are a hint, not a proof. One unequal pair is enough to show it is not arithmetic.
💡Compare this to Topic 1.3: a constant rate of change gives points that lie on a straight line. The graph of an arithmetic sequence is a set of points on a line, and d is that line's slope.
2.1.A2.1.A.3Math
The General Term of an Arithmetic Sequence
Start at the initial value a0 and add d once for every step. After n steps you have added d a total of n times, which gives the general term:
an = a0 + dn
A context often gives you a term other than the first. If you know the kth term, ak, start there. Going from position k to position n takes n − k steps, each adding d:
an = ak + d(n − k)
Example. Suppose a3 = 11 and d = 4. Then an = 11 + 4(n − 3). Distribute: 11 + 4n − 12 = 4n − 1, so an = −1 + 4n and the initial value is a0 = −1. Both forms describe the same sequence.
Set the initial value a0 and the common difference d. Then pick a known term position k and a target position n. The two forms of the general term always give the same value.
initial value a₀3
difference d2
known term k3
target n7
an = a0 + dn = 3 + 2nknown term: a3 = 3 + 2(3) = 9from a0: a7 = 3 + 2(7) = 17from a3: a7 = 9 + 2(7 − 3) = 17✓ Same value both ways.
🔑To find d from two terms ak and am, divide the change in output by the change in input: d = (am − ak) / (m − k). For a2 = 13 and a7 = 33, d = 20 / 5 = 4.
💡When n is smaller than k, the factor n − k is negative: you are stepping backward from ak. That is how you find the initial value from a later term: a0 = ak − dk.
2.1.B2.1.B.1Concept
Geometric Sequences and the Common Ratio
In a geometric sequence, successive terms have a common ratio: each term is r times the term before it. That is constant proportional change. Compare "add 4 each step" with "multiply by 2 each step". They look similar in a list but behave very differently.
r = gn+1 / gn (the same for every n)
The ratio r is…
Positive terms are…
Proportional change
r > 1
increasing
r = 1.08 means 8% growth per step
0 < r < 1
decreasing
r = 0.8 means a 20% decrease per step
Pick a list of terms. The checker divides each term by the one before it. A sequence is geometric only if every one of those ratios is the same.
ratios (each term ÷ the one before): 2, 2, 2, 2✓ Geometric: the common ratio is r = 2 (increasing).
⚠️A constant difference is not a constant ratio. The list 5, 10, 15, 20, 25 adds 5 every time, so it is arithmetic. Its ratios are 2, 1.5, 1.33…, 1.25, which are not constant, so it is not geometric.
💡In this lesson the geometric sequences have r > 0, so they are increasing or decreasing. A negative ratio would make the signs alternate, and the sequence would do neither.
2.1.B2.1.B.2Math
The General Term of a Geometric Sequence
Start at the initial value g0 and multiply by r once for every step. After n steps you have multiplied n times:
gn = g0 · rn
If you know the kth term gk instead, start there. Going from position k to position n takes n − k multiplications:
gn = gk · rn − k
Example. Suppose g2 = 12 and r = 2. Then gn = 12 · 2n − 2. Since 2n − 2 = 2n / 4, this is gn = 3 · 2n, so the initial value is g0 = 3. Check at n = 5: 3 · 32 = 96 and 12 · 23 = 96.
Set the initial value g0 and the common ratio r. Then pick a known term position k and a target position n. Watch how g0rn and gkrn − k always agree.
initial value g₀3
ratio r2
known term k2
target n5
gn = g0rn = 3(2)nknown term: g2 = 3(2)2 = 12from g0: g5 = 3(2)5 = 96from g2: g5 = 12(2)5 − 2 = 96✓ Same value both ways. The sequence is increasing (r > 1).
⚠️The exponent goes on r only. 3 · 2n is not (3 · 2)n = 6n. At n = 5 the first gives 96 and the second gives 7,776.
🔑To find r from two terms, use gm = gk · rm − k. For g3 = 24 and g6 = 192, r3 = 192 / 24 = 8, so r = 2. A quantity that grows 5% each step has r = 1.05, and one that loses 15% each step has r = 0.85.
2.1.B2.1.B.3Concept
Arithmetic vs. Geometric: Who Grows Faster?
Both kinds of sequence can increase, but not in the same way. An increasing arithmetic sequence increases equally with each step. An increasing geometric sequence of positive values increases by a larger amount with each successive step.
n
an = 4 + 4n
added
gn = 4(1.5)n
added
0
4
—
4
—
1
8
+4
6
+2
2
12
+4
9
+3
3
16
+4
13.5
+4.5
4
20
+4
20.25
+6.75
Both sequences start at the same value. The blue one adds d each step (arithmetic). The orange one multiplies by r each step (geometric). Compare how much each one adds.
start value4
difference d4
ratio r1.50
arithmetic adds: 4, 4, 4, 4, 4, …geometric adds: 2, 3, 4.5, 6.75, 10.125, …The arithmetic sequence adds the same amount every step. The geometric sequence adds a larger amount every step.The geometric sequence first becomes larger than the arithmetic one at n = 4.
💡This connects to Topics 1.1 and 1.3. Constant increases mean a constant rate of change, so the points lie on a line. Increases that keep getting larger mean the rate of change is increasing, so the points follow a curve that bends upward (concave up).
⚠️The words "positive values" matter. If g0 = −3 and r = 2, the terms are −3, −6, −12, −24. Each step still changes by a larger amount, but in the negative direction, so the sequence is decreasing. Also, a geometric sequence can start slower and still pass an arithmetic one: in the table, g4 = 20.25 is the first term larger than a4 = 20.