# 11.9.2: Key Equations

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### Key Equations

 Formula for a factorial $0!=1 1!=1 n!=n( n−1 )( n−2 )⋯( 2 )( 1 ), for n≥2 0!=1 1!=1 n!=n( n−1 )( n−2 )⋯( 2 )( 1 ), for n≥2$
 recursive formula for nth term of an arithmetic sequence $a n = a n−1 +d , n≥2 a n = a n−1 +d , n≥2$ explicit formula for nth term of an arithmetic sequence $a n = a 1 +d(n−1) a n = a 1 +d(n−1)$
 recursive formula for $nthnth$ term of a geometric sequence $a n =r a n−1 ,n≥ 2 a n =r a n−1 ,n≥ 2$ explicit formula for $nth nth$ term of a geometric sequence $a n = a 1 r n−1 a n = a 1 r n−1$
 sum of the first $n n$ terms of an arithmetic series $S n = n( a 1 + a n ) 2 S n = n( a 1 + a n ) 2$ sum of the first $n n$ terms of a geometric series $S n = a 1 (1− r n ) 1−r ,r≠1 S n = a 1 (1− r n ) 1−r ,r≠1$ sum of an infinite geometric series with $–1 $S n = a 1 1−r ,r≠1 S n = a 1 1−r ,r≠1$
 number of permutations of $n n$ distinct objects taken $r r$ at a time $P(n,r)= n! (n−r)! P(n,r)= n! (n−r)!$ number of combinations of $n n$ distinct objects taken $r r$ at a time $C(n,r)= n! r!(n−r)! C(n,r)= n! r!(n−r)!$ number of permutations of $n n$ non-distinct objects $n! r 1 ! r 2 !… r k ! n! r 1 ! r 2 !… r k !$
 Binomial Theorem $(x+y) n = ∑ k−0 n ( n k ) x n−k y k (x+y) n = ∑ k−0 n ( n k ) x n−k y k$ $(r+1)th (r+1)th$ term of a binomial expansion $( n r ) x n−r y r ( n r ) x n−r y r$
 probability of an event with equally likely outcomes $P(E)= n(E) n(S) P(E)= n(E) n(S)$ probability of the union of two events $P(E∪F)=P(E)+P(F)−P(E∩F) P(E∪F)=P(E)+P(F)−P(E∩F)$ probability of the union of mutually exclusive events $P(E∪F)=P(E)+P(F) P(E∪F)=P(E)+P(F)$ probability of the complement of an event $P(E')=1−P(E) P(E')=1−P(E)$

11.9.2: Key Equations is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.