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8-10. Solve the word problem involving series and sequence. A frog wants to reach the end of a bamboo tree that has a height of 78 meters. The frog leaps 4 meters during the day and slips 2 meters down during the night. How many days will it take the frog to reach the top of the bamboo?CHAPTER 2 : SEQUENCE AND SERIES Calling a "progression" in British English) is an ordered list of numbers: the The persis of a sunflower by layer. nd also the resulting vale, are called the " sum" or the 'summation" some amongdes ang I. In the puzzle called the Tower of Hanoi, the object is to use a series of moves to take minimum number of moves required to move n rings is " for I ring . 3 for 2 rings , 7 for Tings: 15 for & rings, and an for 5 rings earrings 3. The population of bacteria after a period of time . Lesson 1: Sequence and Series Lesson Proper LINTRODUCTION Sequence and series can be seen almost everywhere in our su miothe me to under with decreasing height with a patiem or a sequence. carded by the DOH. Their data would consist of a sequence. INTERACTION Sequence is a succession of numbers in a specific order. Each number in a sequence . It can be denoted by A. .Usualy when written, a sequence is separate EQUENCE Example: Try to find the next three terms of each infinite sequence. SOLUTION: a. The next term is & less than any precede 1arm are 3. decreased by twice a ano room therapyasfrom the first form to the third term hess means that you have a formula an = 27 - 8(n-1). where n is the set of poshave integers REMEMBER. i an aninmate sequence is a sequence in which each term at nce is obtained from subtracting a term to the preceding term. - FORMULA: ; a - as + (n - 1)d, where d is the common difference and n is the number of terms at term and an is the with term 8 there i ramada inoriginal) actually used the quence A. - As + (n - 1)d, ", -35-8n when we have anotice tomays for the sequence which is a. = 35-Sm. where n is the set of In= 1. then a, =35-8(1) =27 FIRST term n =2, then a, -35-8(2)-19 SECOND term Ifn - 3, then a, =35-8(3) =11 THIRD term e formula to get the 100" term ", =35-8(100) =-765 and perm is 10 times the preceding term. The ms are 500.0, 5,000.0, COCOCOCOCOCO--. . BIG SEQUENCE An geometric sequence is a sequence in which each term after the first is iratio. The common ratio is obtained from dividing a term to the preceding FORMULA: an * dirt, where r is the common ratio To get the formula in finding the nith term of the sequence, we are going to use the Where a, = 0.5 and $10" Part e is left for you to practice on. Just ren 'd above, we have the formula for arithmetic Arithmetic series: S. = (a, + a. ) series: S. = 901-72,ral 9 example: Determine the first five terms of each defined sequence, and a formula for its Therefore, the sequence is arithmetic. To get the and five terms:", - 3-1--2 S. -Ha, +a.) S, - 1(-2+(-2) S, =-10 -3-5--2 The sum of the first five terms of the sequence (ve terms: a, -4(1)+3-7 4 . Therefore, the sequel dor man dinerin pas ", - 4(2)+3-1 get the formula for its series we use To a, =4(3)+3=15 ", - 4(5)+3- 23 34(7+23) The sum of the first five ter 2. =4n+3 is 75. of the sequence UMMATION NOTATION The inconvenience of writing so many terms can be minimized usit used to indicate the " sum' f() denotes some functional expression involving variable i. then Upper bound (+1) This expression is sad as "the lower bound summation of 27 + 1 from i = 010 im Example 1: Wine each senes in ingthe notation AStity+ + + 100 SOLUTION: he expression, which means to be able to find the bound As follows , the last term is fi - so that gives -40+ 20-10+5+.-12 SOLUTION: C. 1+3+5+7+ ..+ 101 SOLUTION: [(2K-1) sterence d = 2. We have, a. =1+(1-1)2 Again we don't know which term is the last term. W 101 - 20-1 La-50 _Always check your answer because for this situation if we use 50 as the upper bound last bern in the preceding example can also be expressed in sigma notation as. 1+3+5+7+ ..+ 101 - [(24 + 1) " Try to test ? this notation is still equivalent to the one in the example. . Basic Properties of Summation 1. call , where c is a constant 11 2 2c, = [a , where e is a constant Example 1: {(2-31)-82-324 -202) -3(1+2+3+4+5+6+7+8+9+10+11 12+13+14 +15 +16+17+18+19+20)