Sequences โ Functions
Arithmetic sequences behave like linear functions ยท Geometric sequences behave like exponential functions
๐ Arithmetic Sequences โ Linear Functions
๐ Arithmetic Sequence
Slope-Intercept Form
aโ = aโ + dยทn
Point-Slope Form
aโ = aโ + dยท(n โ k)
๐ Linear Function
Slope-Intercept Form
f(x) = b + mยทx
Point-Slope Form
f(x) = f(xโ) + mยท(x โ xโ)
๐ข Geometric Sequences โ Exponential Functions
๐ข Geometric Sequence
From term 0
aโ = aโ ยท rโฟ
Point Form
aโ = aโ ยท r^(nโk)
๐ Exponential Function
Standard Form
f(x) = a ยท bหฃ
Point Form
f(x) = f(xโ) ยท b^(x โ xโ)
Linear vs. Exponential โ The Big Difference
How outputs change over equal-length input intervals
How Do Outputs Change Over Equal-Length Input Intervals?
๐ Linear f(x) = b + mx
Change by addition (constant rate)
Outputs increase or decrease by the same amount each time.
Outputs increase or decrease by the same amount each time.
The change d (or slope m) is based on addition.
Test: differences between consecutive outputs are constant.
๐ข Exponential f(x) = aยทbหฃ
Change proportionally (by multiplication)
Outputs are multiplied by the same ratio each time.
Outputs are multiplied by the same ratio each time.
The change r (ratio) is based on multiplication.
Test: ratios between consecutive outputs are constant.
The Two-Point Rule
If you have any two points you can write the equation of a linear function, exponential function, arithmetic sequence, or geometric sequence. Use the point-slope form โ plug in your known point and solve for the rate/ratio.
Identifying from Tables โ Example 1
Check differences (linear) or ratios (exponential) over equal input intervals
How to Identify the Type
Step 1: Confirm x-values are equally spaced (same ฮx each row).
Step 2: Check consecutive output differences. If constant โ Linear.
Step 3: If not constant, check consecutive output ratios. If constant โ Exponential.
Step 4: If neither is constant โ Neither.
a) f(x) Linear
| x | f(x) | ฮ |
|---|---|---|
| 0 | 7 | |
| 3 | 5 | โ2 |
| 6 | 3 | โ2 |
| 9 | 1 | โ2 |
| 12 | โ1 | โ2 |
Linear โ over equal ฮx = 3 intervals, outputs decrease by a constant โ2 each time.
b) g(x) Neither
| x | g(x) | ฮ |
|---|---|---|
| 1 | 0 | |
| 2 | 1 | +1 |
| 3 | 4 | +3 |
| 4 | 9 | +5 |
| 5 | 16 | +7 |
Neither โ differences (1, 3, 5, 7) are not constant, and ratios are also not constant.
c) h(x) Exponential
| x | h(x) | ร |
|---|---|---|
| 0 | 1 | |
| 2 | 2 | ร2 |
| 4 | 4 | ร2 |
| 6 | 8 | ร2 |
| 8 | 16 | ร2 |
Exponential โ over equal ฮx = 2 intervals, outputs are multiplied by a constant ratio of 2.
d) k(x) Exponential
| x | k(x) | ร |
|---|---|---|
| 5 | 80 | |
| 10 | 40 | รท2 |
| 15 | 20 | รท2 |
| 20 | 10 | รท2 |
| 25 | 5 | รท2 |
Exponential (decay) โ over equal ฮx = 5 intervals, outputs are multiplied by a constant ratio of ยฝ.
Building the Equation from Two Points โ Example 2
Geometric sequence (exponential model): rumour spreading
๐ Example 2 โ A rumor spreads geometrically. 43 people heard it on day 3 and 140 people on day 6. How many people have heard the rumor by day 10?
P(3) = 43
P(6) = 140
Find P(10)
1
Use point-slope form for geometric sequences:P(n) = P(3) ยท r^(n โ 3) = 43 ยท r^(n โ 3)
2
Substitute P(6) = 140 to find r:P(6) = 43 ยท r^(6โ3) = 43rยณ = 140
3
Solve for r:rยณ = 140/43 โ r = (140/43)^(1/3)
4
Write the full model:P(n) = 43 ยท (140/43)^((nโ3)/3)
5
Evaluate at n = 10:P(10) = 43 ยท (140/43)^((10โ3)/3) = 43 ยท (140/43)^(7/3) โ 675.58
P(10) โ 676 people have heard the rumor by day 10
Why (n โ 3)/3 as the exponent?
The exponent in the point-slope form is (n โ k) where k is your anchor point (day 3). Since r was found over a span of 3 days, the exponent is (n โ 3)/3 โ NOT just (n โ 3). Each "step" of r covers 3 days, so we divide by 3 to get the right number of steps.
Building the Equation from Two Points โ Example 3
Arithmetic sequence (linear model): theater seats
๐ Example 3 โ Theater rows follow an arithmetic sequence. Row 5 has 31 seats and row 11 has 49 seats. How many seats are in row 25?
s(5) = 31
s(11) = 49
Find s(25)
1
Use point-slope form for arithmetic sequences:s(n) = s(5) + mยท(n โ 5) = 31 + mยท(n โ 5)
2
Substitute s(11) = 49 to find m:s(11) = 31 + mยท(11 โ 5) = 31 + 6m = 49
3
Solve for m:6m = 49 โ 31 = 18 โ m = 18/6 = 3
4
Evaluate at n = 25:s(25) = 31 + 3ยท(25 โ 5) = 31 + 3ยท(20) = 31 + 60 = 91
s(25) = 91 seats in row 25
Quick Reference
Screenshot and save this!
Linear / Arithmetic
f(x) = f(xโ) + mยท(x โ xโ)
Change by addition. Constant differences. Slope m found: m = (yโโyโ)/(xโโxโ)
Exponential / Geometric
f(x) = f(xโ) ยท b^((x โ xโ)/ฮx)
Change by multiplication. Constant ratios. Ratio b found from two known outputs.
Table Test โ Linear
output differences are constant
Subtract consecutive outputs. If you always get the same number โ linear.
Table Test โ Exponential
output ratios are constant
Divide consecutive outputs. If you always get the same ratio โ exponential.
โ ๏ธ Common Mistakes
โ Checking differences on unequal intervals
The input intervals (ฮx) must be equal before testing differences or ratios. Always verify ฮx is constant first.
โ Forgetting to divide exponent by ฮx
In Example 2, r was found over 3 steps, so the exponent is (nโ3)/3. Forgetting to divide by 3 gives the wrong answer.
โ Confusing "neither" with exponential
If differences aren't constant, check ratios before saying "neither". A function must fail BOTH tests to be neither.
โ Using slope formula for sequences
Point-slope form works for both: aโ = aโ + d(nโk) for arithmetic, aโ = aโ ยท r^(nโk) for geometric.