Honors Algebra II: Sequences, Series, & Binomial Expansion Test Review

Honors Algebra II: Sequences and Series Test Review I. Formulas for Sequences and Series Arithmetic Sequence: 𝑎 𝑛 = 𝑎 1 + ( 𝑛 − 1 ) 𝑑 a n ​ =a 1 ​ +(n−1)d 𝑆 𝑛 = 𝑛 2 ( 𝑎 1 + 𝑎 𝑛 ) S n ​ = 2 n ​ (a 1 ​ +a n ​ ) Geometric Sequence: 𝑎 𝑛 = 𝑎 1 ⋅ 𝑟 ( 𝑛 − 1 ) a n ​ =a 1 ​ ⋅r (n−1) Infinite Sum: 𝑆 𝑛 = 𝑎 1 1 − 𝑟 S n ​ = 1−r a 1 ​ ​ for ∣ 𝑟 ∣ < 1 ∣r∣<1 Finite Sum: 𝑆 𝑛 = 𝑎 1 ( 1 − 𝑟 𝑛 ) 1 − 𝑟 S n ​ = 1−r a 1 ​ (1−r n ) ​ II. Writing the First Five Terms of the Sequence Sequence: 𝑎 𝑛 = 4 𝑛 − 3 a n ​ =4n−3 Terms: 1, 5, 9, 13, 17 Arithmetic with 𝑑 = 4 d=4 Sequence: 𝑎 𝑛 = 5 ( − 2 ) 𝑛 − 1 a n ​ =5(−2) n−1 Terms: 5, -10, 20, -40, 80 Geometric with 𝑟 = − 2 r=−2 Sequence: 𝑎 𝑛 = ( 2 𝑛 − 1 ) 𝑛 a n ​ =(2n−1)n Terms: 1, 6, 15, 28, 45 Neither arithmetic nor geometric III. Find the Indicated Term Sequence: 𝑎 𝑛 = 3 𝑛 2 7 − 2 𝑛 a n ​ = 7−2n 3n 2 ​ 𝑎 4 = − 28 a 4 ​ =−28 IV. Finding the Sum Sum of ∑ 𝑖 = 1 5 ( 𝑖 2 + 3 𝑖 ) ∑ i=1 5 ​ (i 2 +3i) 4 + 10 + 18 + 28 + 40 = 100 4+10+18+28+40=100 Sum of ∑ 𝑘 = 0 8 8 ∑ k=0 8 ​ 8 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 48 8+8+8+8+8+8+8+8=48 Sum of ∑ 𝑖 = 0 3 ( − 2 ) 𝑖 ∑ i=0 3 ​ (−2) i 1 − 2 + 4 − 8 = − 5 1−2+4−8=−5 V. First Five Terms and Nature of Sequence Sequence: 𝑎 𝑛 = 150 − 7 𝑛 a n ​ =150−7n Terms: 143, 136, 129, 122, 115 Arithmetic with 𝑑 = − 7 d=−7 Sequence: 𝑎 𝑛 = 8 ( 3 2 ) 𝑛 − 1 a n ​ =8( 2 3 ​ ) n−1 Terms: 8, 12, 18, 27, 81/2 Geometric with 𝑟 = 3 2 r= 2 3 ​ Sequence: 𝑎 𝑛 = 9 , 𝑎 𝑛 + 1 = 𝑎 𝑛 − 13 a n ​ =9,a n+1 ​ =a n ​ −13 Terms: 9, -4, -17, -30, -43 Arithmetic with 𝑑 = − 13 d=−13 Sequence: 𝑎 𝑛 = 4 , 𝑎 𝑛 + 1 = 3 𝑎 𝑛 a n ​ =4,a n+1 ​ =3a n ​ Terms: 4, 12, 36, 108, 324 Geometric with 𝑟 = 3 r=3 VI. Recursive and Explicit Formulas Geometric Sequence: 𝑎 = 5 , 𝑟 = − 4 a=5,r=−4 Explicit: 𝑎 𝑛 = 5 ( − 4 ) 𝑛 − 1 a n ​ =5(−4) n−1 Terms: 5, -20, 80, -320, 1280 Arithmetic Sequence: 𝑎 = 3 7 , 𝑑 = 4 3 a= 7 3 ​ ,d= 3 4 ​ Explicit: 𝑎 𝑛 = 3 7 + ( 𝑛 − 1 ) 4 3 a n ​ = 7 3 ​ +(n−1) 3 4 ​ Arithmetic Sequence: 𝑑 = − 15 d=−15 Explicit: 𝑎 𝑛 = − 15 𝑛 + 22 a n ​ =−15n+22 Geometric Sequence: 𝑟 = − 2 r=−2 Explicit: 𝑎 𝑛 = 6 ( − 2 ) 𝑛 − 1 a n ​ =6(−2) n−1 Geometric Sequence: 𝑟 = 1 2 r= 2 1 ​ Explicit: 𝑎 𝑛 = 4 ( 1 2 ) 𝑛 − 1 a n ​ =4( 2 1 ​ ) n−1 Arithmetic Sequence: 𝑑 = − 55 d=−55 Explicit: 𝑎 𝑛 = 324 − 55 ( 𝑛 − 1 ) a n ​ =324−55(n−1) VII. Indicated Term for Arithmetic Sequence 𝑎 = 2 5 , 𝑑 = 2 , 𝑎 𝑛 = ? a= 5 2 ​ ,d=2,a n ​ =? 𝑎 2 = 12 5 a 2 ​ = 5 12 ​ 𝑎 = 5 , 𝑑 = 8 , 𝑎 21 = ? a=5,d=8,a 21 ​ =? 𝑎 21 = 285 a 21 ​ =285 𝑎 = 103 , 𝑑 = − 7 , 𝑎 15 = ? a=103,d=−7,a 15 ​ =? 𝑎 15 = 68 a 15 ​ =68 VIII. Indicated Term for Geometric Sequence 𝑎 = 5 , 𝑟 = − 2 5 , 𝑎 𝑛 = ? a=5,r= 5 −2 ​ ,a n ​ =? 𝑎 7 = 1 3125 a 7 ​ = 3125 1 ​ 𝑎 = 3 , 𝑟 = − 5 , 𝑎 3 = ? a=3,r=−5,a 3 ​ =? 𝑎 3 = 75 a 3 ​ =75 𝑎 = 640 , 𝑟 = 1 8 , 𝑎 𝑛 = ? a=640,r= 8 1 ​ ,a n ​ =? 𝑎 7 = 5 a 7 ​ =5 IX. Recursive to Explicit Arithmetic: 𝑑 = 9 , 𝑎 = 8 d=9,a=8 Explicit: 𝑎 𝑛 = 9 𝑛 − 1 a n ​ =9n−1 Geometric: 𝑟 = 12 , 𝑎 = 7 ( 12 ) 𝑛 − 1 r=12,a=7(12) n−1 Explicit: 𝑎 𝑛 = 𝑎 1 ( 12 ) 𝑛 − 1 a n ​ =a 1 ​ (12) n−1 Arithmetic: 𝑑 = 7.5 d=7.5 Explicit: 𝑎 𝑛 = 7.5 𝑛 − 5 a n ​ =7.5n−5 Geometric: 𝑟 = − 2 r=−2 Explicit: 𝑎 𝑛 = 50 ( − 2 ) 𝑛 − 1 a n ​ =50(−2) n−1 X. Series Convergence and Sum Arithmetic: Sum of 13 terms 𝑆 13 = 1196 S 13 ​ =1196 Geometric: 𝑛 = 8 n=8 Sum: 𝑆 8 = 76.5 S 8 ​ =76.5 Arithmetic Sum: Finite 𝑆 350 = 243 , 250 S 350 ​ =243,250 Geometric Sum: Convergent 𝑆 8 = 625 S 8 ​ =625 Geometric Sum: Convergent 𝑆 7 = − 39 , 062 S 7 ​ =−39,062 Arithmetic Sum: Convergent 𝑆 8 = 2294 S 8 ​ =2294 XI. Series Convergence Geometric Series Sum: 𝑆 = 21 / 5 S=21/5 Geometric Series Sum: 𝑆 = 21 / 2 S=21/2 Geometric Series: Divergent Arithmetic Series: Divergent XII. Binomial Expansion Expand ( 3 𝑥 − 2 𝑦 ) 7 (3x−2y) 7 using Pascal's Triangle. Find coefficient of 𝑎 2 a 2 term of ( 2 𝑎 + 1 ) 5 (2a+1) 5 . Coefficient: 40 4th term of ( 4 𝑦 + 𝑥 ) 4 (4y+x) 4 . Term: 16 𝑥 𝑦 3 16xy 3 XIII. Piecewise Functions 𝐶 ( 𝑥 ) C(x) defined piecewise. 𝐶 ( 2 ) = 0 C(2)=0 𝐶 ( 7 ) = 20 C(7)=20 𝐶 ( 15 ) = 0 C(15)=0