Number Sense Formula Memorizations

Formulas from Section 2 of Bryant Heath's Number Sense guide

Sum of the first m integers

m(m+1)/2

Sum of the First m Odd Integers

m²

Sum of the First m Even Integers

m(m+1)

Sum of First m Squares

m(m+1)(2m+1)/6

Sum of First m Cubes

(m(m+1)/2)²

1²-2²+3²-4²+5²…

±m(m+1)/2

Sum of a General Arithmetic Series a₁+a₂+…+a_m

m(a₁+a_m)/2

Number of Terms of General Arithmetic Sequence a₁+a₂+…+a_m

a_m-a₁/d +1 where d is the common difference

Sum of an Infinite Geometric Sequence a₁(1+d+d²+…)

a₁/(1-d)

First Fibonacci Number

1

Second Fibonacci Number

1

Third Fibonacci Number

2

Fourth Fibonacci Number

3

Fifth Fibonacci Number

5

Sixth Fibonacci Number

8

Seventh Fibonacci Number

13

Eighth Fibonacci Number

21

Ninth Fibonacci Number

34

Tenth Fibonacci Number

55

Eleventh Fibonacci Number

89

Twelfth Fibonacci Number

144

Thirteenth Fibonacci Number

233

Fourteenth Fibonacci Number

377

Fifteenth Fibonacci Number

610

Sum of the First n Fibonacci Numbers

F_n+2-1

Sum of First 3 Numbers in a Fibonacci Sequence

2a+2b

Sum of First 4 Numbers in a Fibonacci Sequence

3a+4b

Sum of First 5 Numbers in a Fibonacci Sequence

5a+7b

Sum of First 6 Numbers in a Fibonacci Sequence

8a+12b

Sum of First 7 Numbers in a Fibonacci Sequence

13a+20b

Sum of First 8 Numbers in a Fibonacci Sequence

21a+33b

Sum of First 9 Numbers in a Fibonacci Sequence

34a+54b

Sum of First 10 Numbers in a Fibonacci Sequence

55a+88b

Sum of First 11 Numbers in a Fibonacci Sequence

89a+143b

Sum of First 12 Numbers in a Fibonacci Sequence

144a+232b

Number of Divisors (the prime factorization is p₁^e₁*p₂^e₂*…p_n^e_n)

(e₁+1)(e₂+1)…(e_n+1)

Sum of Divisors (the prime factorization is p₁^e₁*p₂^e₂*…p_n^e_n)

(p₁^e₁+1-1)/(p₁-1) * (p₂^e₂+1-1)/(p₂-1)…(p_n^e_n+1-1)/(p_n-1)

Number of Relatively Prime Integers less than n

(p₁-1)/p₁ * (p₂-1)/p₂ * (p_n-1)/p_n * n

Sum of Relatively Prime Integers less than n

(Number of Relatively Prime Integers)*n/2

Number of Diagonals for an n-gon

n(n-3)/2

Sum of Exterior Angles

360

Exterior Angle for n-gon

360/n

Interior Angle

180(n-2)/n

Sum of Interior Angles

180(n-2)

The nth M-gonal number

n[(M-2)*n-(M-4)]/2

Sum of Two Consecutive Triangular Numbers (T_n-1+T_n)

n²

Sum of First n Triangular Numbers

m(m+1)(m+2)/6

Sum of the Same Triangular and Pentagonal Numbers (T_n+P_n)

2n²

Triangle Sides for Acute

a²+b²>c²

Triangle Sides for Obtuse

a²+b²<c²

Right Triangle Sides When Only Given One Odd Side (ex. the sides of a right triangle are integers, one of its sides is 11, what is the other side/hypotenuse)

The integers right before and after n²/2 (ex. 11²/2=60.5 so 60/61)

Right Triangle Sides When Only Given One Odd Side (ex. the sides of a right triangle are integers, one of its sides is 10, what is the other side/hypotenuse)

Divide until you get to an odd number, then scale the triangle up (ex. 5²/2=12.5 so 12/13, then multiply by 2 to get 24/26)

Area of an Equilateral Triangle when Knowing the Side Length

s²*sqrt(3)/4

Area of an Equilateral Triangle when Knowing the Height

h²*sqrt(3)/3

Finding the Height of an Equilateral Triangle when Given the Side Length

s*sqrt(3)/2

Cube Surface Area where s is a Side Length

6s²

Sphere Volume

(4/3)pi*r³

Sphere Surface Area

4*pi*r²

Cone Volume

(1/3)pi*r²*h

Cone Surface Area

pi*r*(slant height)+pi*r²

Cylinder Volume

pi*r²*h

Cylinder Surface Area

2pi*r*h

Face Diagonal of a Cube

s*sqrt(2)

Body Diagonal of a Cube

s*sqrt(3)

nCk/nPk=

1/(k!)

nPk/nCk=

k!

Pythagorean (sin/cos)

sin^2+cos^2=1

Pythagorean (cot/csc)

1+cot^2=csc^2

Pythagorean (tan/sec)

tan^2+1=sec^2

sin(a+-b)

sin(a)cos(b)+-sin(b)cos(a)

cos(a+-b)

cos(a)cos(b)-+sin(a)sin(b)

sin(2a)

2sin(a)cos(a)

cos(2a)

cos^2(a)−sin^2(a)

cos(2a) (only sin)

1−2sin^2(a)

cos(2a) (only cos)

2cos^2(a)−1

sin(90−θ)

cos(θ)