STEM High School Prealculus Week 6: Sequences and Series, Exams of Political studies

Download STEM High School Prealculus Week 6: Sequences and Series and more Exams Political studies in PDF only on Docsity! COMMONWEALTH HIGH SCHOOL Ecols St., Commonwealth, Quezon City SENIOR HIGH SCHOOL DEPARTMENT Science, Technology, Engineering and Mathematics (STEM) Strand PREALCULUS WEEK 6 SEQUENCES AND SERIES MOST ESSENTIAL LEARNING COMPETENCIES At the end of the module, the students should be able to: illustrate a series; [STEM_PC11SMI-Ih-1] differentiate a series from a sequence; [STEM_PC11SMI-Ih-2] use the sigma notation to represent a series; and [STEM_PC11SMI-Ih-3] apply the use of sigma notation in finding sums. -RONr Definition A sequence is a succession of numbers that follows a pattern. An infinite sequence is a function whose domain is the set of consecutive positive integers. If the domain is the set of the first n positive integers, then it is said to be a finite sequence. The individual elements in a sequence are called terms. A series is the sum of nth terms of a sequence. REVIEW OF SEQUENCE AND SERIES ARITHMETIC SEQUENCE AND SERIES An arithmetic sequence is a sequence where the difference between two consecutive terms is constant. That is, each term is obtained by adding a nonzero constant, called common difference, to its preceding term. The nth term of an arithmetic sequence is given by, a, =a,+(n—-1)d, Arithmetic series is the sum S, ofa finite arithmetic sequence is given by n Ss = pla +n) n where a, is the nth term; n a, is the first term; and dis the common difference GEOMETRIC SEQUENCE AND SERIES A geometric sequence is a sequence where the second and the succeeding terms are obtained by multiplying the preceding term by a nonzero constant called common ratio. The nth term of a geometric sequence is given by n-1 a, = ar Geometric series is the sum S, of a finite geometric sequence is given by a, (1- r" ) S, =——— where r 41 1l-r where a, is the nth term; n a, is the first term; and ris the common difference SIGMA NOTATION The sigma notation is defined by the equation qd DG =A, + Ogg + Ang ++ Ag, +A, Ep where p and q are integers such that p< g. The number p is called the lower limit of the sum while q is called the upper limit. Examples: Represent the following series in sigma notation. 1. 3+6+9+12+15 Solution: We have already established in example 2.1.2a that the sequence whose terms are given above has the general term a, =3n. The terms 3, 6, 9, 12, 15 are obtained by replacing n with 1, 2, 3, 4, 5, respectively. Hence, 5 34+64+94+124+15= 31. isl 2. 4+94+164+25+36 Solution: The given terms are the squares of consecutive integers 2, 3, 4, 5, and 6. Hence, the general term of the sequence containing these terms is a, =n’. Thus, 6 44+94+164+25+36=S'7. i= 3B. X+2x7 432° 44x" +...4. 9x17 Solution: X+ 2X9 439 447 4...4 9x17 = i i(x?**) 9 i=1 Y self Check 1 Represent the following series in sigma notation. 1, 242° +27+2°+94498 ext x Q. + 4+—4+...45= 2 3 4 45 3. 1-3+5-74+9-11+...-29 Examples: Determine the sum of the following. 3 1. x3" i=l Solution: 3" = 3! 4 324 4 33H i=l = 37 +3°+3% =94+27+81 =117 5 2. Y.2i+5 i=l Solution: S:2i-+5=(2(1) +5) (2(2) +5) +(2(3) +5)+(2(4) +5)-+(2(5) +5) =74+9+4114+134+15 =55 ACTIVITY 1 Determine the sum of each sigma notation. 8 lL. S3n-1 n=1 a. 97 c. 99 b. 98 d. 100 2 yn ‘ — a. 0.49 c. 0.84 b. 0.69 d. 1.02 S ni( Qn 3. -1)"" zc) a ¢, 42 © 415 “17 17 a 9 60 * 20 3 4. y2(3y" n=1 a, 182 ¢, 127 "27 "82 b 27 d 82 182 "127 4 n 5. Line a, 41 c, 40 ‘51 ' 67 p, SL a 7 “41 * 40 ACTIVITY 2 Solve each problem and choose the letter of the correct answer. 1. How many consecutive integers beginning with 1 would you have to add in order to get 2080? a. 60 c. 64 b. 62 d. 66 2. Find the sum of the first 30 terms of an arithmetic sequence 4, 11, 18, 25, ... a. 3160 c. 3170 b. 3165 d. 3175 3. An auditorium has a 15 rows, with 20 seats in the front row and 2 more seats in each row thereafter. How many seats are there in all? a. 510 c. 530 b. 520 d. 540 4. In the arithmetic sequence 5, 9, 13, ... , 37, what is the index of 37? a. 7 c. 9 b. 8 d. 10 5. Given the arithmetic sequence 5, 9, 13, ... , 37, find the sum of the first 60 terms. a. 7380 c. 7580 b. 7480 d. 7680 ACTIVITY 3 Solve each problem and choose the letter of the correct answer. 1. What is the 7% term of 2, 1 2 3° 2’ 3” 2 a. 2 wlaAr b. aINaln Mr Ed Delos Reyes willed one-half of his state to his eldest child, one-half of the remainder to his second child and so on until his 4 child who is the youngest received P1.25 million. What was the total value of the estate? a. P18.25 million c. P10 million b. P20 million d. P6.26 million What is the fifth term of the geometric sequence with first term 1024 and second term 512? a. 256 c. 64 b. 128 d. 32 Find the sum of the infinite geometric sequence with third term 3 and common _ 2 ratio —. 3 5 a. = c= 5 3 b. 3 d.5 Suppose that JM accepts a job that pays a salary of P150,000 the first year with a yearly increase of 5%. How much will her salary be after 10 years? a. P232,699 c. P676,399 b. P323,799 d. P767,299 SUMMATIVE QUIZ Solve each problem and choose the letter of the correct answer. 1. Given the sequence 3, 4, 7, 9, 11, , what is the next term? a. 12 c. 16 b. 13 d. 15 Given the sequence -1,-2 Hs; 0, , what is the next term? a 1 c 1 ut b. = 6 Given the sequence 1, 1, 2, 3, 5, 8, , what is the next term? a. 5 c 1 b. 8 d. 13 Given the sequence 100, 126, 152, , what is the next term? a. 178 c. 198 b. 230 d. 278 Given the sequence i uv vo 2 , what is the next term? 4° 12° 12° 4 35 35 a, -— Cc — 12 2 2 12 b. = d. = 3 35 If a,=5 and a, = : , what is a, of this geometric sequence? 1 5 a. = ec = 16 16 b. L d. 5 8 8 If a, =12 and a, =972, then what is a, of this geometric sequence? a. 1944 c. 729 b. 972 d. 8748 8. If a, =5 and a,=125, then what is a, of this geometric sequence? a. 390,625 1 1 “ 390,625 78,125 d. 78,125 9. If a, = 3 and a; = 81 , then what is a,, of this geometric sequence? 16 a 1 «19,683 " 262,144 * 262,144 p, 19,683 q, 19,683 512 2 10. If a, = 1024 and a, = 262,144 | then what is a,? 3 3 3 c. 4 Jt a. 4 3 3 11. Find S,, of the sequence a, = 2+(n-1)3. a. 155 c. 170 b. 110 d. 160 12. Find S,, of the sequence a, =-5+2(n-1). a. 80 c. 40 b. 50 d. 100 13. Find S,, of the sequence a, = 7+(n-1)4. a. 240 c. 260 b. 250 d. 270 14. Find S, of the sequence a, =3(2)". a. -3 c. —93 b. 186 d. 93 15. What is S, of the sequence a, =3(2)"? a. 6 c. 93 b. 2 d. cannot be determined