a. Answer recursive. Explanation Given: MULTI-STEP The table shows several terms of a sequence. Calculation: Since a 2 − a 1 = 25 − ( − 12 ) = 37 a n d a 3 − a 2 = − 49 − 25 = − 74 ≠ 37 , so the sequence does not have common difference, so it is not arithmetic. Since a 2 a 1 = 25 − 12 = − 25 12 a n d a 3 a 2 = − 49 25 ≠ − 25 12 , so there is no common ratio .so it is not geometric. Note however that if we multiply each term by -2 and then add 1, we get the next term since ⇒ − 2 ( − 12 ) + 1 = 24 + 1 = 25 , ⇒ − 2 ( 25 ) + 1 = − 50 + 1 = − 49 , and ⇒ − 2 ( − 49 ) + 1 = 98 + 1 = 99. Since each term then depends on the previous term, it can be written as a recursive sequence. b. To determine To find: The next term in the sequence. Answer Next term of pattern is -197. Explanation Given: The table shows several terms of a sequence. Calculation: From part (a), the pattern is multiply by -2 and then add 1, we get the next term is , - 2 ( 99 ) + 1 = − 198 + 1 = − 197 Next term of pattern is -197. c. To determine To find: The formula which can describe the sequence. Answer a 1 = − 12 , a n = − 2 a n − 1 + 1 , n ≥ 2 . Explanation Given: The table shows several terms of a sequence. Calculation: we multiply each term by -2 and then add 1, we get the next term since ⇒ − 2 ( − 12 ) + 1 = 24 + 1 = 25 , ⇒ − 2 ( 25 ) + 1 = − 50 + 1 = − 49 , and ⇒ − 2 ( − 49 ) + 1 = 98 + 1 = 99. Since each term then depends on the previous term, it can be written as a recursive sequence. a 1 = − 12 , a n = − 2 a n − 1 + 1 , n ≥ 2 . The correct answer is then choice B. Conclusion: d. To determine To explain: Whether an explicit formula be written as a linear equation for the sequence or not. Answer No. Explanation Given: The table shows several terms of a sequence. Calculation: For an explicit formula of a sequence to be written as a linear equation, the sequence must be arithmetic since both athematic sequence and linear function need a constant rate of change (a constant difference), we know this sequence is not arithmetic so it cannot be written as a linear equation.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN: 9780079039897

Author: Carter

Publisher: McGraw Hill

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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

Chapter 7.10, Problem 36PFA

To determine

To identify: The type of sequence.

Expert Solution

Answer to Problem 36PFA

recursive.

Explanation of Solution

Given: MULTI-STEP The table shows several terms of a sequence.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018, Chapter 7.10, Problem 36PFA , additional homework tip 1

Calculation:

Since a2−a1=25−(−12)=37 and a3−a2=−49−25=−74≠37 , so the sequence does not have common difference, so it is not arithmetic.

Since a2a1=25−12=−2512 and a3a2=−4925≠−2512 , so there is no common ratio .so it is not geometric.

Note however that if we multiply each term by -2 and then add 1,

we get the next term since

⇒−2(−12) + 1=24+1=25, ⇒−2(25) +1= −50+1=−49, and⇒−2(−49)+1=98+1=99.

Since each term then depends on the previous term, it can be written as a recursive sequence.

To determine

To find: The next term in the sequence.

Expert Solution

Answer to Problem 36PFA

Next term of pattern is -197.

Explanation of Solution

Given: The table shows several terms of a sequence.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018, Chapter 7.10, Problem 36PFA , additional homework tip 2

Calculation:

From part (a), the pattern is multiply by -2 and then add 1, we get the next term is ,

- 2(99)+1=−198+1=−197

Next term of pattern is -197.

To determine

To find: The formula which can describe the sequence.

Expert Solution

Answer to Problem 36PFA

a1=−12 , an=−2an−1+1 , n≥2 .

Explanation of Solution

Given: The table shows several terms of a sequence.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018, Chapter 7.10, Problem 36PFA , additional homework tip 3

Calculation:

we multiply each term by -2 and then add 1, we get the next term since

⇒−2(−12) + 1=24+1=25, ⇒−2(25) +1= −50+1=−49, and⇒−2(−49)+1=98+1=99.

Since each term then depends on the previous term, it can be written as a recursive sequence.

a1=−12 , an=−2an−1+1 , n≥2 .

The correct answer is then choice B.

Conclusion:

To determine

To explain: Whether an explicit formula be written as a linear equation for the sequence or not.

Expert Solution

Answer to Problem 36PFA

No.

Explanation of Solution

Given: The table shows several terms of a sequence.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018, Chapter 7.10, Problem 36PFA , additional homework tip 4

Calculation:

For an explicit formula of a sequence to be written as a linear equation, the sequence must be arithmetic since both athematic sequence and linear function need a constant rate of change (a constant difference), we know this sequence is not arithmetic so it cannot be written as a linear equation.

Chapter 7.10, Problem 35HP

Chapter 7.10, Problem 37PFA

Chapter 7 Solutions

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

Ch. 7.1 - Prob. 1AGP Ch. 7.1 - Prob. 1BGP Ch. 7.1 - Prob. 1CGP Ch. 7.1 - Prob. 1DGP Ch. 7.1 - Prob. 2AGP Ch. 7.1 - Prob. 2BGP Ch. 7.1 - Prob. 3GP Ch. 7.1 - Prob. 4AGP Ch. 7.1 - Prob. 4BGP Ch. 7.1 - Prob. 5GP

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