LEC 4 poitngh gutgi lyj jjb fffff kunf.pptx

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1. Lecture no 4 Sequences and Summations

2. A sequence is a function from a subset of the set of integers (usually either the set {0 , 1 , 2 , . . . } or the set {1 , 2 , 3 , . . . }) to a set S . We use the notation to denote the image of the integer n . We call a term of the sequence. 1, 2, 3, 5, 8… 1, 3, 9, 27… EXAMPLE 1 Consider the sequence { }, where = 1/n The list of the terms of this sequence, beginning with , namely, , , , , . . . , starts with 1 , 1/2 ,1/3,1/4…  

3. A geometric progression is a sequence of the form where the initial term a and the common ratio r are real numbers. EXAMPLE 2 The sequences { } with = , { } with = , and { } with = are geometric progressions with initial term and common ratio equal to 1 and −1; 2 and 5; and 6 and 1 / 3, respectively, if we start at n = 0. The list of terms begins with 1 , −1 , 1 , −1 , 1 , . . . ; the list of terms begins with 2 , 10 , 50 , 250 , 1250 , . . . ; and the list of terms begins with 6, 2,2/3,2/9,2/27, . . .  

4. An arithmetic progression is a sequence of the form a, a + d, a + 2 d, . . . , a + nd, . . . where the initial term a and the common difference d are real number. EXAMPLE 3 The sequences { } with = −1 + 4 n and { } with = 7 − 3 n are both arithmetic progressions with initial terms and common differences equal to −1 and 4, and 7 and −3, respectively, if we start at n = 0. The list of terms , , , , . . . begins with −1 , 3 , 7 , 11 , . . . , and the list of terms , , , , . . . begins with 7 , 4 , 1 , −2 , . . . .  

5. A recurrence relation for the sequence { } is an equation that expresses in terms of one or more of the previous terms of the sequence, namely, , , . . . , , for all integers n with n ≥ , where is a nonnegative integer. A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation. It relates the index n with the term value. EXAMPLE 5 Let { } be a sequence that satisfies the recurrence relation = + 3 for n = 1 , 2 , 3 , . . . , and suppose that = 2. What are , , and ? Solution: We see from the recurrence relation that = + 3 = 2 + 3 = 5. It then follows that = 5 + 3 = 8 and = 8 + 3 = 11 . EXAMPLE 6 Let { } be a sequence that satisfies the recurrence relation = − for n =2 , 3 , 4 , . . . , and suppose that = 3 and = 5. What are and ? Solution: We see from the recurrence relation that = − = 5 − 3 = 2 and = − = 2 − 5 = −3.We can find , , and each successive term in a similar way  

6. The Fibonacci sequence , , is defined by the initial conditions = 0 , = 1, and the recurrence relation = + for n = 2 , 3 , 4 , . . . . It is also used to calculate the growth of rabbit population. EXAMPLE 7 Find the Fibonacci numbers = + = 1 + 0 = 1 , = + = 1 + 1 = 2 , = + = 2 + 1 = 3 , = + = 3 + 2 = 5 , = + = 5 + 3 = 8 .  

7. Suppose that { an } is the sequence of integers defined by = n !, the value of the factorial function at the integer n , where n = 1 , 2 , 3 , . . . is n ! = n((n − 1 )(n − 2 ) . . . 2.1 ) = n(n − 1 ) ! = , The sequence of factorials satisfies the recurrence relation =  

8. EXAMPLE 11 Compound Interest Suppose that a person deposits $10,000 in a savings account at a bank yielding 11% per year with interest compounded annually. How much will be in the account after 30 years? Solution: To solve this problem, let denote the amount in the account after n years. Because the amount in the account after n years equals the amount in the account after n −1 years plus interest for the n th year, we see that the sequence { } satisfies the recurrence relation = + 0 . 11 = ( 1 . 11 ) . The initial condition is = 10 , 000. We can use an iterative approach to find a formula for .  

9. Note that = ( 1 . 11 ) = ( 1 . 11 ) = ( 1 . 11 ) 2 = ( 1 . 11 ) = ( 1 . 11 ) 3 ... = ( 1 . 11 ) = ( 1 . 11 )n . When we insert the initial condition = 10 , 000, the formula = ( 1 . 11 )n 10 , 000 is obtained. Inserting n = 30 into the formula = ( 1 . 11 )n 10 , 000 shows that after 30 years the account contains = ( 1 . 11 ) 3010 , 000 = $228 , 922 . 97 .  

10. Use of sequence by a Con-Man A king and a con-man Story Once upon a time, in a faraway kingdom, there lived a wise king known for his love of intellectual challenges. One day, a clever con artist arrived at the palace, claiming he had invented a new game that was both entertaining and educational. He presented the game of chess to the king, who was instantly captivated. The king, thrilled by the depth and strategy involved, wanted to reward the con artist. "Name your reward, and it shall be yours," the king said, confident in his kingdom's vast wealth. The con artist, with a humble bow, made a seemingly modest request. " Your Majesty, I ask for just a few grains of rice. Place one grain of rice on the first square of the chessboard, two grains on the second, four on the third, and so on, doubling the amount with each square until all 64 squares are filled ."The king chuckled at the simplicity of the request and ordered his servants to fulfill it immediately. But as they began the task, the king soon realized the enormity of what had been asked. By the time they reached the 20th square, the amount of rice required was already over a million grains. By the 40th square, the number had reached the trillions, and by the final square, the total was an unimaginable quantity – more than all the rice in the entire kingdom, and far more than could ever be produced on Earth.

11. 1 , 2 , 4 , 8 , 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472, 274877906944, 549755813888, 1099511627776, 2199023255552, 4398046511104, 8796093022208, 17592186044416, 35184372088832, 70368744177664, 140737488355328, 281474976710656, 562949953421312, 1125899906842624, 2251799813685248, 4503599627370496, 9007199254740992, 18014398509481984, 36028797018963968, 72057594037927936, 144115188075855872, 288230376151711744, 576460752303423488, 1152921504606846976, 2305843009213693952, 4611686018427387904, 9223372036854775808 . Use of sequence by a Con-Man A king and a con-man Story

12. The global production of rice is vast. According to the Food and Agriculture Organization (FAO), about 740 million metric tons of rice are produced globally each year (as of recent data).To convert this into grains of rice, it’s important to note that the weight of a single grain of rice varies slightly depending on the type of rice. On average, one grain of rice weighs around 0.02 grams (20 milligrams).Let's calculate the approximate number of grains of rice produced annually: 1 metric ton = 1,000,000 grams 740 million metric tons = 740,000,000,000,000 grams (740 trillion grams) Now, dividing this by the weight of a single grain (0.02 grams) :740,000,000,000,000 grams 0.02 grams per grain=37,000,000,000,000,000 grains 0.02 grams per grain740,000,000,000,000 grams​ =37,000,000,000,000,000 grains So, approximately 37 quadrillion grains of rice are produced globally each year. The number 9,223,372,036,854,775,808 can be written in words as: Nine quintillion, two hundred twenty-three quadrillion, three hundred seventy-two trillion, thirty-six billion, eight hundred fifty-four million, seven hundred seventy-five thousand, eight hundred eight.

13. Summations Let we have a sequence then the sum of its term can be denoted by + +…+ . j is called the index of summation. It acts as a dummny variable m is lower limit and n is upper limit. EXAMPLE 17 Use summation notation to express the sum of the first 100 terms of the sequence { }, where Solution: The lower limit for the index of summation is 1, and the upper limit is 100.We write this sum as  

14. EXAMPLE 18 What is the value of ? Solution: = = 1 + 4 + 9 + 16 + 25 = 55 . EXAMPLE 19 What is the value of Solution: = = 1 + ( −1 ) + 1 + ( −1 ) + 1 = 1 .