8.3 ARITHMETIC AND GEOMETRIC SEQUENCES

1 Pg451 [R] G1 5-36058 / HCG / Cannon & Elich cr 11-30-95 MP1. Airthmetic and GEOMETRIC SEQUENCES 451. ARITHMETIC AND GEOMETRIC SEQUENCES . Whenever you tell me that mathematics is just a human invention like the game of chess I would like to believe you. But I keep returning to the same problem. Why does the mathematics we have discovered in the past so often turn out to describe the workings of the Universe? John Barrow I remember that when I Two kinds of regular SEQUENCES occur so often that they have specific names, was about twelve I learned ARITHMETIC and GEOMETRIC SEQUENCES . We treat them together because some obvi- from [my uncle] that by the ous parallels between these kinds of SEQUENCES lead to similar formulas. This also distributive law 21 times makes it easier to learn and work with the formulas.

2 The greatest value in this 21 equals 11. I thought that was great. association is understanding how the ideas are related and how to derive the Peter Lax formulas from fundamental concepts. Anyone learning the formulas this way can recover them whenever needed. Both ARITHMETIC and GEOMETRIC SEQUENCES begin with an arbitrary first term, and the SEQUENCES are generated by regularly adding the same number (the com- mon difference in an ARITHMETIC sequence) or multiplying by the same number (the common ratio in a GEOMETRIC sequence). Definitions emphasize the parallel fea- tures, which examples will clarify. Definition: ARITHMETIC and GEOMETRIC SEQUENCES ARITHMETIC Sequence a1 5 a and an 5 an21 1 d for n . 1. The sequence $an% is an ARITHMETIC sequence with first term a and common difference d.

3 GEOMETRIC Sequence a1 5 a and an 5 r an21 for n . 1. The sequence $an% is a GEOMETRIC sequence with first term a and common ratio r. The definitions imply convenient formulas for the nth term of both kinds of SEQUENCES . For an ARITHMETIC sequence we get the nth term by adding d to the first term n 2 1 times; for a GEOMETRIC sequence, we multiply the first term by r, n 2 1. times. Formulas for the nth terms of ARITHMETIC and GEOMETRIC SEQUENCES For an ARITHMETIC sequence, a formula for the nth term of the sequence is an 5 a 1 ~n 2 1!d. (1). For a GEOMETRIC sequence, a formula for the nth term of the sequence is an 5 a r n21. (2). The definitions allow us to recognize both ARITHMETIC and GEOMETRIC SEQUENCES . In an ARITHMETIC sequence the difference between successive terms, an11 2 an , is always the same, the constant d; in a GEOMETRIC sequence the ratio of successive an11.

4 Terms, , is always the same. an pg452 [V] G2 5-36058 / HCG / Cannon & Elich kr 11-20-95 QC1. 452 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers cEXAMPLE 1 ARITHMETIC or GEOMETRIC ? The first three terms of a se- quence are given. Determine if the sequence could be ARITHMETIC or GEOMETRIC . If it is an ARITHMETIC sequence, find d; for a GEOMETRIC sequence, find r. 1 1 1. (a) 2, 4, 8, .. (b) ln 2, ln 4, ln 8, .. (c) , , ,.. 2 3 4. Strategy: Calculate the dif- ferences and /or ratios of Solution successive terms. (a) a2 2 a1 5 4 2 2 5 2, and a3 2 a2 5 8 2 4 5 4. Since the differences are a2 4. not the same, the sequence cannot be ARITHMETIC . Checking ratios, 5 5 2, a1 2. a 8. and 3 5 5 2, so the sequence could be GEOMETRIC , with a common ratio a2 4. r 5 2. Without a formula for the general term, we cannot say anything more about the sequence.

5 (b) a2 2 a1 5 ln 4 2 ln 2 5 ln~ 42 ! 5 ln 2, and a3 2 a2 5 ln 8 2 ln 4 5. ln~ 84 ! 5 ln 2, so the sequence could be ARITHMETIC , with ln 2 as the common difference. As in part (a), we cannot say more because no general term is given. (c) a2 2 a1 5 13 2 12 5 2 16 , and a3 2 a2 5 14 2 13 5 2 121 . The differences are a2 ~ 13 ! 2. not the same, so the sequence is not ARITHMETIC . 5 5 , and a1 ~ 12 ! 3. a3 ~ 14 ! 3. 5 5 , so the sequence is not GEOMETRIC . Note that the sequence in part a2 ~ 13 ! 4. (a) could be GEOMETRIC and the sequence in part (b) could be ARITHMETIC , but in part (c) you can conclude unequivocally that the sequence cannot be either ARITHMETIC or GEOMETRIC . b cEXAMPLE 2 ARITHMETIC or GEOMETRIC ? Determine whether the sequence is ARITHMETIC , GEOMETRIC , or neither. (a) $3 2 (b) $2n% (c) an 5 ln n Solution (a) a2 2 a1 5 ~3 2 2!

6 2 ~3 2 1! 5 ~ ! 2 5 , and a3 2 a2 5 ~3 2 3! 2 ~3 2 2! 5 From the first three terms, this could be an ARITHMETIC sequence with d 5 Check the difference an11 2 an . an11 2 an 5 @3 2 ~n 1 1!# 2 @3 2 # 5 The sequence is ARITHMETIC , with d 5 (b) a2 2 a1 5 4 2 2 5 2, and a3 2 a2 5 8 2 4 5 4, so the sequence is not ARITHMETIC . Using the formula for the general term, an11 2n11. 5 n 5 2. an 2. The sequence $2n% is GEOMETRIC , with 2 as the common ratio. pg453 [R] G1 5-36058 / HCG / Cannon & Elich kr 11-20-95 QC1. Airthmetic and GEOMETRIC SEQUENCES 453. n11. (c) an11 2 an 5 ln~n 1 1! 2 ln n 5 ln . The difference depends on n, so n a ln~n 1 1! the sequence is not ARITHMETIC . Checking ratios, n11 5 , so the ratio an ln n also changes with n. The sequence is neither ARITHMETIC nor GEOMETRIC . b cEXAMPLE 3 ARITHMETIC SEQUENCES Show that the sequence is ARITHMETIC .

7 Find the common difference and the twentieth term. (a) an 5 2n 2 1 (b) 50, 45, 40, .. , 55 2 5n, .. Solution (a) The first few terms of $an% are 1, 3, 5, 7, .. , from which it is apparent that each term is 2 more than the preceding term; this is an ARITHMETIC sequence with first term and common difference a 5 1 and d 5 2. Check to see that an11 2 an 5 2. To find a20 , use either the defining formula for the sequence or Equation (1) for the nth term: a20 5 2 20 2 1 5 39 or a20 5 a 1 19d 5 1 1 19 2 5 39. (b) If bn 5 55 2 5n, then bn11 2 bn 5 @55 2 5~n 1 1!# 2 @55 2 5n# 5 25. This is an ARITHMETIC sequence with a 5 50, d 5 25, and so b20 5 55 2. 5 20 5 245. b Given the structure of ARITHMETIC and GEOMETRIC SEQUENCES , any two terms completely determine the sequence. Using Equation (1) or (2), two terms of the sequence give us a pair of equations from which we can find the first term and either the common difference or common ratio, as illustrated in the next example.

8 CEXAMPLE 4 ARITHMETIC SEQUENCES Suppose $an% is an ARITHMETIC se- quence with a8 5 6 and a12 5 24. Find a, d, and the three terms between a8. and a12 . Solution From Equation (1), a8 5 a 1 7d, and a12 5 a 1 11d, from which the difference is given by a12 2 a8 5 4d. Use the given values for a8 and a12 to get 24 2 6 5 4d, or d 5 2 52 . Substitute 2 52 for d in 6 5 a 1 7d and solve for a, a 5 472 . Find the three terms between a8 and a12 by successively adding 2 52 : 5 7 5 5 3. a9 5 a8 2 5 , a10 5 a9 2 5 1, a11 5 a10 2 52 . 2 2 2 2 2. Therefore, a9 is 72 , a10 is 1, and a11 is 2 32 . b cEXAMPLE 5 GEOMETRIC SEQUENCES Determine whether the sequence is GEOMETRIC . If it is GEOMETRIC , then find the common ratio and the terms a1 , a3 , and a10 . (a) $2n%. 2 2. (b) 2, 2 , , .. , 2 2. 3 9. S D. 1. 3.

9 N21. ,.. pg454 [V] G2 5-36058 / HCG / Cannon & Elich kr 11-20-95 QC1. 454 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers Solution Strategy: The property that (a) The first few terms are 2, 4, 8, 16, .. , each of which is twice the preceding identifies a GEOMETRIC se- term. This is a GEOMETRIC sequence with first term a 5 2, and common ratio quence is the common ratio: a given by r 5 n11 an 5 2n 5 2. Using an 5 2 , 2n11 n a a a the values 2 , 3 , 4 , .. a1 a2 a3. must all be the same. For a a1 5 2 a3 5 23 5 8 and a10 5 210 5 1024. GEOMETRIC sequence, use Equation (2). (b) Consider the ratio an11. 5. S D. 2 2. 1. 3. n 1. 52 , an S D. 2 2. 1. 3. n21 3. so the sequence is GEOMETRIC with a 5 2 and r 5 2 13 . Using an 5 2~2 13 !n21, we get a1 5 2, a3 5 ar 2 5 29 , and a10 5 ar 9 5 2~2 13 !

10 9 5 2 19683. 2.. b Partial Sums of ARITHMETIC SEQUENCES There is a charming story told about Carl Freidrich Gauss, one of the greatest mathematicians of all time. Early in Gauss' school career, the schoolmaster as- signed the class the task of summing the first hundred positive integers, 1 1 2 1. 3 1 1 99 1 100. That should have occupied a good portion of the morning, but while other class members busied themselves at their slates calculating 1 1. 2 5 3, 3 1 3 5 6, 6 1 4 5 10, and so on, Gauss sat quietly for a few moments, wrote a single number on his slate, and presented it to the teacher. Young Gauss observed that 1 and 100 add up to 101, as do the pair 2 and 99, 3 and 98, and so on up to 50 and 51. There are fifty such pairs, each with a sum of 101, for a total of 50 101 5 5050, the number he wrote on his slate.