Algebra I · Algebra II · Algebra I–II · Quick reference
Algebra1 · COLLEGE
Grades 8–12 · College6 pages
Real numbers, exponents and radicals, equations and inequalities, polynomials and factoring, rational expressions, lines and systems, quadratics and complex numbers, functions, exponentials and logs, sequences, conics and word problems.
1Real numbers2Exponents3Linear eq.4Polynomials5Rational6Lines & systems7Word problems8Quadratics9Conics10Functions11Sequences
1Real Numbers & Operationsthe number system, its laws, and one agreed order
- ∈ / ∉
- is / is not an element of: 3 ∈ ℤ, ½ ∉ ℤ
- ⊂
- is a subset of: ℕ ⊂ ℤ
- ∪ / ∩
- union (in either) / intersection (in both)
- ∅
- the empty set, { }
- {x | x > 2}
- set-builder: “all x such that x > 2”
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | a + b = b + a | ab = ba |
| Associative | (a + b) + c = a + (b + c) | (ab)c = a(bc) |
| Identity | a + 0 = a | a · 1 = a |
| Inverse | a + (−a) = 0 | a · 1a = 1, a ≠ 0 |
| Distributive | a(b + c) = ab + ac |
Zero product: if ab = 0, then a = 0 or b = 0. Division by 0 is undefined.
Commuters move, associates regroup
Commutative: the numbers commute, trading places (order changes). Associative: the same numbers change who they associate with (parentheses move).
Signed numbers
- Add, same signs: add, keep the sign. −4 + (−7) = −11
- Add, different signs: subtract, keep the sign of the larger |n|. −9 + 4 = −5
- Subtract = add the opposite: 5 − (−3) = 5 + 3 = 8
- × and ÷: same signs → +, different → −. (−6)(−4) = 24; −18 ÷ 3 = −6
- An even count of negative factors gives +; odd gives −.
- |−7| = 7: absolute value is distance from 0.
Order of operations
P parentheses & other groupings ( ) [ ] fraction bars, radicals · E exponents · MD multiply/divide, left → right · AS add/subtract, left → right
Please Excuse My Dear Aunt Sally
The traditional PEMDAS line. M and D are partners (whichever comes first, left to right); so are A and S. 18 ÷ 3 × 2 is 12, not 3.
Order of operations Worked
3 + 4 × 5 = 3 + 20 = 23multiply first
18 ÷ 3 × 2 = 6 × 2 = 12left → right
2[5 + (8 − 2)²] = 2[5 + 36] = 82inside out
20 − 2³ ÷ 4 + 1 = 20 − 2 + 1 = 19
12 + 89 − 5 = 204 = 5bar groups
Fractions
- ab + cd = ad + bcbd: 23 + 14 = 8 + 312 = 1112
- ab · cd = acbd; to divide, multiply by the reciprocal: 34 ÷ 98 = 34 · 89 = 23
Keep · Change · Flip
To divide fractions: keep the first, change ÷ to ×, flip the second.
| Words | Algebra |
|---|---|
| sum, more than, increased by | x + 4 |
| 5 less than x | x − 5 (order flips!) |
| product, of, times | 3x |
| quotient, per, ratio | x2 |
| twice, double / half | 2x / x2 |
| is, equals, results in | = |
| at least / at most | ≥ / ≤ |
| ½ | ¼ | ¾ | ⅕ | ⅛ | ⅓ | |
|---|---|---|---|---|---|---|
| decimal | 0.5 | 0.25 | 0.75 | 0.2 | 0.125 | 0.3̄ |
| percent | 50% | 25% | 75% | 20% | 12.5% | 33⅓% |
| Divisible by | if… |
|---|---|
| 2 | last digit even |
| 3 | digit sum divisible by 3 (471: 12 ✓) |
| 4 | last two digits divisible by 4 |
| 5 | ends in 0 or 5 |
| 6 | divisible by 2 and 3 |
| 9 | digit sum divisible by 9 |
| 10 | ends in 0 |
GCF and LCM by primes Worked
24 = 2³ · 3 and 36 = 2² · 3²
GCF: lowest powers of shared primes = 2² · 3 = 12
LCM: highest powers of all primes = 2³ · 3² = 72
check: 12 · 72 = 24 · 36 = 864 ✓
Ratios, proportions & percent
- ab = cd ⟺ ad = bc (cross-multiply)
- part = percent × whole: 15% of 60 = 0.15 × 60 = 9
- 42 is what % of 56? 4256 = 0.75 = 75%
- Properties of equality: add, subtract, multiply or divide both sides by the same number (not 0 for ÷) and the equation stays true.
Mental math is algebra
- (a − b)(a + b) = a² − b²: 49 · 51 = 50² − 1² = 2,499
- (10n + 5)² = 100n(n + 1) + 25: 65² → 6 · 7 = 42, then 25 → 4,225
- a(b − c) = ab − ac: 7 × 98 = 700 − 14 = 686
- x% of y = y% of x: 8% of 25 = 25% of 8 = 2
- × 11 (two digits): 36 × 11 → 3 | 3+6 | 6 = 396
Averages · estimating roots Worked
mean = sumcount; scores 82, 90, 85 → sum 257
4th score for a mean of 88: 88 · 4 − 257 = 95need = target × n − sum
√50 lies between 7 and 8 (49 < 50 < 64), near 7: ≈ 7.07estimate roots
2Exponents & Radicalsa ≠ 0 throughout
| Law | Rule | Example |
|---|---|---|
| Product | am · an = am+n | 2³ · 2⁴ = 2⁷ = 128 |
| Quotient | aman = am−n | 5652 = 5⁴ = 625 |
| Power of a power | (am)n = amn | (3²)⁴ = 3⁸ = 6,561 |
| Power of a product | (ab)n = anbn | (2x³)² = 4x⁶ |
| Power of a quotient | (ab)n = anbn | (23)³ = 827 |
| Zero | a⁰ = 1 | 7⁰ = 1 |
| Negative | a−n = 1an | 2−3 = 18 |
| Rational | am/n = (ⁿ√a)m | 82/3 = 2² = 4 |
One rung down the ladder
Same base: multiply the powers → add exponents; divide → subtract; power of a power → multiply. Each operation drops one rung: ^ → ×, × → +, ÷ → −.
Simplify with the laws Worked
(2x³y)² · x⁻⁴ = 4x⁶y² · x⁻⁴ = 4x²y²
12a5b24a2b5 = 3a3b3subtract
Sign & sum traps
- −3² = −9 but (−3)² = 9: the exponent touches only what is beside it.
- (a + b)² ≠ a² + b²: (3 + 4)² = 49, not 25.
- 2−3 is positive: 18, not −8.
Scientific notation
c × 10n with 1 ≤ c < 10.
- 4,500,000 = 4.5 × 10⁶ · 0.0032 = 3.2 × 10−3
- (3 × 10⁴)(2 × 10⁵) = 6 × 10⁹
- 8 × 1074 × 10−2 = 2 × 10⁹
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
| n³ | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 | 1331 | 1728 |
Radical rules (a, b ≥ 0)
- √(ab) = √a · √b; √(ab) = √a√b
- √(x²) = |x|; an even root of a negative is not real
- ⁿ√(am) = am/n
- ∛(−8) = −2 (odd roots of negatives are real); ⁴√(−16) is not real
Simplify & combine Worked
√72 = √(36 · 2) = 6√2largest square
∛54 = ∛(27 · 2) = 3∛2
√12 + √27 = 2√3 + 3√3 = 5√3like radicals
(2 + √3)(2 − √3) = 4 − 3 = 1conjugates
Rationalize the denominator Worked
6√3 · √3√3 = 6√33 = 2√3
43 − √5 · 3 + √53 + √5 = 4(3 + √5)4 = 3 + √5conjugate
Radical equation: check! Worked
√(x + 3) = x − 3
x + 3 = x² − 6x + 9 → x² − 7x + 6 = 0square both sides
x = 1 or x = 6
x = 1: √4 = 2 but 1 − 3 = −2 ✗ · x = 6: 3 = 3 ✓ → x = 6
3Linear Equations & Inequalitiesdo the same thing to both sides
Solving a first-degree equation
1. Clear fractions (× LCD) · 2. Distribute · 3. Combine like terms · 4. Variables to one side · 5. Isolate · 6. Check in the original.
Solve Worked
3(x − 4) + 2 = 5x − 18
3x − 10 = 5x − 18 → 8 = 2x → x = 4
Check: 3(0) + 2 = 2; 20 − 18 = 2 ✓
x3 + x4 = 7 → 4x + 3x = 84 → x = 12× 12
2x + 1 = 2x + 5 → 1 = 5: no solution
3(x + 2) = 3x + 6: all realsidentity
Literal equations (solve for a letter) Worked
A = ½bh → h = 2Ab
F = 95C + 32 → C = 59(F − 32)F = 212 → C = 100
Clear decimals first Worked
0.2x + 1.5 = 3.1 → ×10 → 2x + 15 = 31 → x = 8
Inequalities
Same steps, but multiplying or dividing by a negative flips the sign.
EX−3x + 5 > 14 → −3x > 9 → x < −3
Flip the sign, flip the sign
Multiply or divide both sides by a negative sign? Then flip the inequality sign. (−2 < 3, but 2 > −3.)
| Inequality | Interval | Graph |
|---|---|---|
| x > 2 | (2, ∞) | open dot → |
| x ≤ −1 | (−∞, −1] | closed dot ← |
| −1 ≤ x < 3 | [−1, 3) | closed · open |
| all reals | (−∞, ∞) | whole line |
Parentheses = open dot (not included); brackets = closed dot. ∞ always gets a parenthesis.
Compound inequalities Worked
−4 < 2x + 2 ≤ 8 → −6 < 2x ≤ 6 → −3 < x ≤ 3and: between
x < −1 or x ≥ 4 → (−∞, −1) ∪ [4, ∞)or: union
Absolute value
- |x − 3| = 5 → x − 3 = 5 or −5 → x = 8 or −2
- |A| < c → −c < A < c: |2x − 1| ≤ 7 → −3 ≤ x ≤ 4
- |A| > c → A < −c or A > c: |x + 2| > 3 → x < −5 or x > 1
- |A| = negative → no solution
Less thAND, greatOR
|A| less than c joins into one and interval (between). |A| greater than c splits into two pieces joined by or.
4Polynomials & Factoringmultiply out · factor back
- Term
- a number, a variable, or their product: 4x³
- Degree
- highest exponent: 4x³ − 2x + 7 has degree 3
- Leading coef.
- coefficient of the highest term (4)
- Mono/bi/trinomial
- 1 / 2 / 3 terms
- Standard form
- terms in descending powers
Add & subtract: combine like terms Worked
(3x² + 2x − 5) − (x² − 4x + 1)
= 3x² + 2x − 5 − x² + 4x − 1 = 2x² + 6x − 6distribute the −
FOIL
First · Outer · Inner · Last for two binomials: (2x − 3)(x + 4) = 2x² + 8x − 3x − 12 = 2x² + 5x − 12. For bigger products, use the box.
Special products
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
- (a ± b)³ = a³ ± 3a²b + 3ab² ± b³
Dividing polynomials
Long division (any divisor) Worked
(2x² + 3x − 5) ÷ (x + 2)
2x² ÷ x = 2x; 2x(x + 2) = 2x² + 4x; subtract → −x − 5
−x ÷ x = −1; −1(x + 2) = −x − 2; subtract → −3
= 2x − 1 − 3x + 2remainder −3
Rational root theorem
Any rational root of a polynomial with integer coefficients is ± factor of constantfactor of leading coef.. For x³ − 6x² + 11x − 6, try ±1, ±2, ±3, ±6: 1, 2 and 3 all work.
Synthetic division (divisor x − r)
- Write r and the coefficients (0 for any missing power).
- Bring down; multiply by r; add down the column; repeat.
- Last number = remainder = f(r) (remainder theorem). Remainder 0 → (x − r) is a factor.
Factoring: a step-by-step path
- 1. GCF first: 6x³ + 9x² = 3x²(2x + 3)
- 2. Count terms. Two → squares or cubes. Three → trinomial. Four → grouping.
- 3. Factor each factor again until nothing factors.
GCF goes first
Before any pattern, pull out the Greatest Common Factor. It makes every later step smaller.
Trinomials Worked
x² + 7x + 12: two numbers × to 12, + to 7 → 3, 4
= (x + 3)(x + 4)
6x² + 7x − 3: ac = −18 → 9 and −2ac method
6x² + 9x − 2x − 3 = 3x(2x + 3) − 1(2x + 3)split, group
= (3x − 1)(2x + 3)
Binomials
- a² − b² = (a − b)(a + b): 9x² − 25 = (3x − 5)(3x + 5)
- a² + b² does not factor over the reals.
- a³ ± b³ = (a ± b)(a² ∓ ab + b²): x³ − 8 = (x − 2)(x² + 2x + 4)
SOAP for cubes
Signs in a³ ± b³ = (a ○ b)(a² ○ ab ○ b²): Same, Opposite, Always Positive.
Grouping (four terms) Worked
x³ + 3x² − 4x − 12 = x²(x + 3) − 4(x + 3)
= (x + 3)(x² − 4) = (x + 3)(x − 2)(x + 2)factor again
Solve by factoring Worked
2x² = 8x → 2x² − 8x = 0never ÷ x
2x(x − 4) = 0 → x = 0 or 4zero product
5Rational Expressionsfractions with polynomials
Domain first
PQ is undefined where Q = 0. x + 2x² − 9: x ≠ 3, −3. Keep these restrictions through every step.
Simplify: factor, then cancel factors Worked
x² − 9x² + 5x + 6 = (x − 3)(x + 3)(x + 2)(x + 3)
= x − 3x + 2, x ≠ −3, −2
Negative exponents become fractions Worked
x⁻¹ + y⁻¹ = 1x + 1y = x + yxy
Only factors may leave
Cancel factors (things multiplied), never terms (things added). x + 55 is not x: the 5 on top is a term.
Multiply, divide, add Worked
x3 · 6x² = 6x3x² = 2x
x² − 14 ÷ x + 12 = (x − 1)(x + 1)4 · 2x + 1 = x − 12flip
3x + 2x + 1 = 3(x + 1) + 2xx(x + 1) = 5x + 3x(x + 1)LCD x(x + 1)
Complex fraction Worked
1 + 1/x1 − 1/x² · x²x² = x² + xx² − 1× LCD x²
= x(x + 1)(x − 1)(x + 1) = xx − 1
Rational equations: check for extraneous roots Worked
2x − 1 = 3x + 1 → 2x + 2 = 3x − 3 → x = 5cross-multiply
xx − 3 = 3x − 3 + 2 → x = 3 + 2(x − 3) → x = 3× (x − 3)
x = 3 makes a denominator 0 → no solutionextraneous
6Lines, Graphs & Systemsslope is rise over run
Slope
(1, 3) and (4, 9): m = 9 − 34 − 1 = 63 = 2
m = y2 − y1x2 − x1 = riserun
HOY-VUX
Horizontal line: slope 0, equation y = c. Vertical line: slope Undefined, equation x = c.
| Form | Equation | Best when you know |
|---|---|---|
| Slope-intercept | y = mx + b | slope and y-intercept |
| Point-slope | y − y1 = m(x − x1) | slope and a point |
| Standard | Ax + By = C | intercepts; systems |
Line through (1, 3) and (4, 9) Worked
m = 2 → y − 3 = 2(x − 1)point-slope
y = 2x + 1slope-intercept
2x − y = −1standard
x-int: −½; y-int: 1set y = 0, x = 0
Parallel & perpendicular
- Parallel: same slope.
- Perpendicular: slopes are negative reciprocals (product −1): 2 and −½.
- EX ⊥ to y = 2x + 1 through (2, 5): y − 5 = −½(x − 2) → y = −½x + 6
Systems of two equations
Substitution Worked
y = 2x + 1 and x + y = 7
x + (2x + 1) = 7 → 3x = 6 → x = 2replace y
y = 2(2) + 1 = 5 → (2, 5)
Elimination Worked
3x + 2y = 16 and x − 2y = 0
add: 4x = 16 → x = 4; then 4 − 2y = 0 → y = 2
(4, 2); check 12 + 4 = 16 ✓
| Lines | Slopes | Solutions |
|---|---|---|
| cross | different | one |
| parallel | same, different b | none |
| same line | same, same b | infinitely many |
Lonely letter → substitute
If a variable is already alone (y = …), substitute. If coefficients match or are opposites, eliminate.
7Word Problemsname it · frame it · solve it · check it
Name it, frame it, solve it, check it
Name the unknown (let x = what is asked). Frame an equation from the facts. Solve. Check that the answer fits the story, with units.
| Type | Model |
|---|---|
| Distance | d = rt |
| Work | 1a + 1b = 1t |
| Mixture | amount × rate, before = after |
| Simple interest | I = Prt |
| Percent change | new − oldold × 100% |
| Direct / inverse | y = kx / y = kx |
| Joint | y = kxz |
Distance Worked
Trains leave opposite ways at 50 and 70 mph. When are they 300 mi apart?
50t + 70t = 300 → t = 2.5 h
Work Worked
One pump fills a tank in 3 h, another in 6 h. Together?
13 + 16 = 1t → 12 = 1t → 2 h
Mixture Worked
Liters of 40% acid to add to 10 L of 10% for 20%?
0.10(10) + 0.40x = 0.20(10 + x) → x = 5 L
Interest · percent · variation Worked
$2,000 at 5% simple for 3 yr: I = 2000(0.05)(3) = $300
80 → 92: 1280 = 15% increase
y varies inversely with x; y = 6 at x = 4 → k = 24; x = 8 → y = 3
Integers & proportions Worked
Three consecutive odd integers sum to 75: x + (x + 2) + (x + 4) = 75 → 23, 25, 27
3 cups per 4 servings → 10 servings: 34 = c10 → 7.5 cups
Tickets Worked
30 sold: adults $8, kids $5, total $189
8a + 5(30 − a) = 189 → 13 adults, 17 kids
8Quadratics & Complex Numbersax² + bx + c, a ≠ 0
| Form | Shows |
|---|---|
| y = ax² + bx + c | y-intercept c; vertex x = −b2a |
| y = a(x − h)² + k | vertex (h, k) |
| y = a(x − r)(x − s) | roots r, s |
Four ways to solve Worked
Factor: x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → 2, 3
Square root: (x − 2)² = 9 → x − 2 = ±3 → 5, −1
Complete the square: x² + 6x + 1 = 0 → (x + 3)² = 8 → −3 ± 2√2+9 both sides
Formula: 2x² + 3x − 2 = 0 → −3 ± √254 → ½, −2
Quadratic formula
x = −b ± √(b² − 4ac)2a
Sing it
To the public-domain tune Pop Goes the Weasel: “x equals negative b, plus or minus the square root, of b squared minus four a c, all over two a.”
Half it, square it, add it
To complete the square on x² + bx: take half of b, square it, add it to both sides. x² + 6x → (6/2)² = 9 → (x + 3)².
| b² − 4ac | Roots | Graph |
|---|---|---|
| > 0, perfect square | 2 rational | crosses x-axis twice |
| > 0, not a square | 2 irrational | crosses twice |
| = 0 | 1 (double) | touches once |
| < 0 | 2 complex | misses the axis |
a tells the shape
- a > 0 opens up (minimum); a < 0 opens down (maximum).
- |a| > 1 narrower; |a| < 1 wider.
Height of a thrown ball Worked
h = −16t² + 64t (feet, seconds)
vertex t = −642(−16) = 2 → h = 64 ftmax
lands: −16t(t − 4) = 0 → t = 4 s
Complex numbers
- i = √(−1), i² = −1; a + bi (a real part, b imaginary part)
- √(−16) = 4i · conjugate of a + bi is a − bi
- x² + 4x + 13 = 0 → −4 ± √(−36)2 = −2 ± 3i
Operate like binomials Worked
(3 + 2i) + (1 − 5i) = 4 − 3i
(2 + 3i)(1 − i) = 2 + i − 3i² = 5 + ii² = −1
1 + i1 − i · 1 + i1 + i = 2i2 = iconjugate
i runs in a circle of 4
i, −1, −i, 1, then repeat. For iⁿ divide n by 4 and use the remainder: i²³ → remainder 3 → −i.
9Conic Sectionsslices of a cone
| Conic | Standard form | Key facts |
|---|---|---|
| Circle | (x − h)² + (y − k)² = r² | center (h, k), radius r |
| Parabola | y − k = a(x − h)² | vertex (h, k); or x − h = a(y − k)² |
| Ellipse | (x − h)²a² + (y − k)²b² = 1 | center (h, k); half-axes a, b |
| Hyperbola | (x − h)²a² − (y − k)²b² = 1 | asymptotes y − k = ±ba(x − h) |
Circle from general form Worked
x² + y² − 4x + 6y − 12 = 0
(x² − 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9complete both squares
(x − 2)² + (y + 3)² = 25: center (2, −3), r = 5
Plus hugs, minus splits
Two squared terms added (same signs) → a closed curve: circle if the denominators match, ellipse if not. Subtracted → a hyperbola, split in two. Only one squared variable → parabola.
Which conic? Ax² + Cy² + Dx + Ey + F = 0
- A = C → circle · A, C same sign → ellipse
- A, C opposite signs → hyperbola · A or C = 0 → parabola
- Foci: ellipse c² = a² − b²; hyperbola c² = a² + b². EX x²9 + y²4 = 1: c = √5
10Functions, Exponentials & Logsone input, exactly one output
- Function
- each input x has exactly one output f(x)
- Domain
- allowed inputs
- Range
- outputs produced
- Vertical line test
- a graph is a function if no vertical line hits it twice
Evaluate f(x) = x² − 3x Worked
f(−2) = 4 + 6 = 10
f(a + 1) = a² + 2a + 1 − 3a − 3 = a² − a − 2
Finding the domain
- No zero denominators: g(x) = 1x + 1 → x ≠ −1
- No negatives under even roots: √(x − 4) → [4, ∞)
- Logs need positive inputs: ln(x − 2) → (2, ∞)
Combine & compose (f = 2x + 1, g = x²) Worked
(f + g)(x) = x² + 2x + 1
(f ∘ g)(3) = f(9) = 19inside first
(g ∘ f)(3) = g(7) = 49order matters
Inverse: swap x and y, solve Worked
y = 2x + 1 → x = 2y + 1 → f−1(x) = x − 12
check f(f−1(5)) = 2(2) + 1 = 5 ✓
Average rate of change
f(b) − f(a)b − a, the slope between two points of the graph.
EXf(x) = x² from 1 to 4: 16 − 13 = 5
Inverse facts
- Exists (as a function) only if f is one-to-one: horizontal line test.
- Graph of f−1 = graph of f reflected in y = x; domain and range trade places.
Moving graphs: y = a·f(x − h) + k
- h: right h · k: up k
- |a| > 1 stretch; a < 0 reflect over the x-axis
- Even: f(−x) = f(x), y-axis symmetry. Odd: f(−x) = −f(x), origin symmetry.
Exponential & logarithmic
- f(x) = a · bˣ: growth if b > 1, decay if 0 < b < 1
- Compound interest A = P(1 + rn)nt; continuous A = Pert
- EX$1,000 at 6% monthly, 5 yr: 1000(1.005)⁶⁰ ≈ $1,348.85
Same-base equations Worked
4x + 1 = 8 → 22x + 2 = 2³
2x + 2 = 3 → x = ½match exponents
Logarithms
- logb x = y means by = x: log₂ 32 = 5, log 1000 = 3, ln e = 1
- log(MN) = log M + log N · log(MN) = log M − log N · log Mᵖ = p log M
- Change of base: logb x = ln xln b
- EX3ˣ = 20 → x = ln 20ln 3 ≈ 2.727
A log is an exponent
log₂ 32 asks “2 to what power is 32?” Answer: 5. Loop it: the base goes to the answer’s power and lands on the inside.
Rational functions: asymptotes
- Vertical: zeros of the denominator (after canceling).
- Horizontal: compare degrees of top (n) and bottom (d).
- EX2x + 1x − 3: VA x = 3, HA y = 2
BOBO · BOTN · EATS-DC
Bigger On Bottom → y = 0. Bigger On Top → None. Exponents Are The Same → Divide Coefficients.
Half-life Worked
80 g, half-life 5 yr; after 15 yr:
80 · (½)15/5 = 80 · ⅛ = 10 g
11Sequences & Seriespatterns and their sums
| Arithmetic (add d) | Geometric (× r) | |
|---|---|---|
| nth term | an = a1 + (n − 1)d | an = a1rn−1 |
| Sum of n | Sn = n(a1 + an)2 | Sn = a1(1 − rn)1 − r |
| Infinite sum | diverges (d ≠ 0) | a11 − r if |r| < 1 |
Arithmetic: 3, 7, 11, … Worked
d = 4: a₂₀ = 3 + 19 · 4 = 79
S₂₀ = 20(3 + 79)2 = 820
Geometric: 2, 6, 18, … Worked
r = 3: a₆ = 2 · 3⁵ = 486
S₆ = 2(1 − 36)1 − 3 = 728
8 + 4 + 2 + … = 81 − ½ = 16r = ½
Pair the ends
1 + 2 + … + 100: pair 1 with 100, 2 with 99, …: 50 pairs of 101 = 5,050. That pairing is the arithmetic-sum formula.
Sigma & binomials
- Σk=14 k² = 1 + 4 + 9 + 16 = 30
- Pascal row 3: 1 3 3 1 → (x + 2)³ = x³ + 6x² + 12x + 8
- n! = n(n − 1)⋯1; 5! = 120; 0! = 1
Recursive form
a₁ = 3, aₙ = aₙ₋₁ + 4 → 3, 7, 11, 15, … (each term from the one before)