Unit 1

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Lesson 1: Sequence (数列)

N-th term a_n

Arithmetic sequence (等差数列) (linear in nature)

Common difference (公差) d=a_(i+1)−a_i

General term (通项公式)

Explicit rule: a_n=a_1+d(n−1)

Recursive rule: an=a(n−1)+d

Geometric sequence (exponential in nature)

Common ration (公比) r=a_(i+1)/a_i

General term (通项公式)

Explicit rule: a_n=a_1⋅r^(n−1)

Recursive rule: an=a(n−1)⋅r

Lesson 2: Series (级数)

Sum of n terms (前n项和)

Arithmetic series

S_n=n/2 (a_1+a_n )=n/2 (2a_1+d(n−1))

Geometric series

Finite: S_n=(a_1 (1−r^n ))/(1−r)

Infinite: S_n=a_1/(1−r), when |r|<1

Annuities

A(t)=P(1+r/n)^nt

Lesson 3: Factorials, Sigma Notation and Recursion

Factorial

n!=n⋅(n−1)⋅…⋅3⋅2⋅1

0!=1

Sigma notation

Lesson 4: Quadratic sequences

T_n=an^2+bn+c

Second difference is constant

(T(i+2)−T(i+1) )−(T_(i+1)−T_i )=2a

Lesson 5: Recursively defined sequences in context

Fix point

Lesson 6: Pascal's triangle and the binomial theorem

(a+b)^n
=_n C_0⋅a^n⋅b^0 +_n C_1 〖⋅a〗^(n−1)⋅b^1+…+_n C_r⋅a^r⋅b^(n−r)+…+_n C_n⋅a^0⋅b^n

=〖Σ(r=0)^n〗( n) C_r⋅a^r⋅b^(n−r)

Questions: Find coefficient or constant terms. Identify a and b in the question

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