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Unit 1 - Coggle Diagram
Unit 1
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 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: a
n=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: a
n=a
(n−1)⋅r
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
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
PreCalc Index