EBK PRECALCULUS W/LIMITS
EBK PRECALCULUS W/LIMITS
4th Edition
ISBN: 9781337516853
Author: Larson
Publisher: CENGAGE CO
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Chapter 9.4, Problem 60E
To determine

To decide whether the sequence 2, 13, 38, 73, 118, 173 , … can be represented perfectly be a linear or a quadratic model and then find a model.

Expert Solution & Answer
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Answer to Problem 60E

The sequence 2, 13, 38, 73, 118, 173 , … can be represented perfectly be a quadratic model an=5n27 .

Explanation of Solution

Given:

Let the sequence be a1=2, a2=13, a3=38, a4=73, a5=118, a6=173, 

Concept used:

  • The first differences of a sequence are found by subtracting consecutive terms. The second differences are found by subtracting consecutive first differences.
  • When the first differences are all the same, the sequence has a perfect linear model. That is, the sequence is arithmetic represented as an=a1+(n1)d .
  • When the second differences are all the same the sequence has a perfect quadratic model which is represented as an=an2+bn+c .

Calculation:

Let’s first find out the first difference:

  T1=a2a1=13(2)=15T2=a3a2=3813=25T3=a4a3=7338=35T4=a5a4=11873=45T5=a6a5=173118=55

And so on.

Since, the first differences are different, now find out second differences.

  T2T1=2515=10T3T2=3525=10T4T3=4535=10T5T4=5545=10

And so on.

The sequence can be represented perfectly by a quadratic model an=an2+bn+c , since, second differences are same.

To find out the model, solve the system of equations:

For n=1,

  a1=a+b+c=2 … (1)

For n=2,

  a2=4a+2b+c=13 … (2)

For n=3,

  a3=9a+3b+c=38 … (3)

Multiply equation (1) with 1 and then add it to the equation (2) and (3).

  abc=24a+2b+c=13_(4)3a+b=15

And,

  abc=29a+3b+c=38_(5)8a+2b=40

Now multiply equation (4) with 2 , and then add it with equation (5).

  6a2b=308a+2b=40_(6)2a=10

Now solve equation (6) for a and get, a=5 .

Now substitute a=4 in equation (4) and solve for b, and get

  3(5)+b=15b=0

Finally substitute a=5 and b=0 in equation (1) and get,

  a+b+c=25+c=2c=7

Therefore, the sequence can be represented perfectly by a quadratic model an=5n27 .

Chapter 9 Solutions

EBK PRECALCULUS W/LIMITS

Ch. 9.1 - Prob. 11ECh. 9.1 - Prob. 12ECh. 9.1 - Prob. 13ECh. 9.1 - Prob. 14ECh. 9.1 - Prob. 15ECh. 9.1 - Prob. 16ECh. 9.1 - Prob. 17ECh. 9.1 - Prob. 18ECh. 9.1 - Prob. 19ECh. 9.1 - Prob. 20ECh. 9.1 - Prob. 21ECh. 9.1 - Prob. 22ECh. 9.1 - Prob. 23ECh. 9.1 - Prob. 24ECh. 9.1 - Prob. 25ECh. 9.1 - Prob. 26ECh. 9.1 - Prob. 27ECh. 9.1 - Prob. 28ECh. 9.1 - Prob. 29ECh. 9.1 - Prob. 30ECh. 9.1 - Prob. 31ECh. 9.1 - Prob. 32ECh. 9.1 - Prob. 33ECh. 9.1 - Prob. 34ECh. 9.1 - Prob. 35ECh. 9.1 - Prob. 36ECh. 9.1 - Prob. 37ECh. 9.1 - Prob. 38ECh. 9.1 - Prob. 39ECh. 9.1 - Prob. 40ECh. 9.1 - Prob. 41ECh. 9.1 - Prob. 42ECh. 9.1 - Prob. 43ECh. 9.1 - Prob. 44ECh. 9.1 - Prob. 45ECh. 9.1 - Prob. 46ECh. 9.1 - Prob. 47ECh. 9.1 - Prob. 48ECh. 9.1 - Prob. 49ECh. 9.1 - Prob. 50ECh. 9.1 - Prob. 51ECh. 9.1 - Prob. 52ECh. 9.1 - Prob. 53ECh. 9.1 - Prob. 54ECh. 9.1 - Prob. 55ECh. 9.1 - Prob. 56ECh. 9.1 - Prob. 57ECh. 9.1 - Prob. 58ECh. 9.1 - Prob. 59ECh. 9.1 - Prob. 60ECh. 9.1 - Prob. 61ECh. 9.1 - Prob. 62ECh. 9.1 - Prob. 63ECh. 9.1 - 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Prob. 31RECh. 9 - Prob. 32RECh. 9 - Prob. 33RECh. 9 - Prob. 34RECh. 9 - Prob. 35RECh. 9 - Prob. 36RECh. 9 - Prob. 37RECh. 9 - Prob. 38RECh. 9 - Prob. 39RECh. 9 - Prob. 40RECh. 9 - Prob. 41RECh. 9 - Prob. 42RECh. 9 - Prob. 43RECh. 9 - Prob. 44RECh. 9 - Prob. 45RECh. 9 - Prob. 46RECh. 9 - Prob. 47RECh. 9 - Prob. 48RECh. 9 - Prob. 49RECh. 9 - Prob. 50RECh. 9 - Prob. 51RECh. 9 - Prob. 52RECh. 9 - Prob. 53RECh. 9 - Prob. 54RECh. 9 - Prob. 55RECh. 9 - Prob. 56RECh. 9 - Prob. 57RECh. 9 - Prob. 58RECh. 9 - Prob. 59RECh. 9 - Prob. 60RECh. 9 - Prob. 61RECh. 9 - Prob. 62RECh. 9 - Prob. 63RECh. 9 - Prob. 64RECh. 9 - Prob. 65RECh. 9 - Prob. 66RECh. 9 - Prob. 67RECh. 9 - Prob. 68RECh. 9 - Prob. 69RECh. 9 - Prob. 70RECh. 9 - Prob. 71RECh. 9 - Prob. 72RECh. 9 - Prob. 73RECh. 9 - Prob. 74RECh. 9 - Prob. 75RECh. 9 - Prob. 76RECh. 9 - Prob. 77RECh. 9 - Prob. 78RECh. 9 - Prob. 79RECh. 9 - Prob. 80RECh. 9 - Prob. 81RECh. 9 - Prob. 82RECh. 9 - Prob. 83RECh. 9 - Prob. 84RECh. 9 - Prob. 85RECh. 9 - Prob. 86RECh. 9 - 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Prob. 15CLTCh. 9 - Prob. 16CLTCh. 9 - Prob. 17CLTCh. 9 - Prob. 18CLTCh. 9 - Prob. 19CLTCh. 9 - Prob. 20CLTCh. 9 - Prob. 21CLTCh. 9 - Prob. 22CLTCh. 9 - Prob. 23CLTCh. 9 - Prob. 24CLTCh. 9 - Prob. 25CLTCh. 9 - Prob. 26CLTCh. 9 - Prob. 27CLTCh. 9 - Prob. 28CLTCh. 9 - Prob. 29CLTCh. 9 - Prob. 30CLTCh. 9 - Prob. 31CLTCh. 9 - Prob. 32CLTCh. 9 - Prob. 33CLTCh. 9 - Prob. 34CLTCh. 9 - Prob. 35CLTCh. 9 - Prob. 36CLTCh. 9 - Prob. 37CLTCh. 9 - Prob. 38CLTCh. 9 - Prob. 39CLTCh. 9 - Prob. 40CLTCh. 9 - Prob. 41CLTCh. 9 - Prob. 1PSCh. 9 - Prob. 2PSCh. 9 - Prob. 3PSCh. 9 - Prob. 4PSCh. 9 - Prob. 5PSCh. 9 - Prob. 6PSCh. 9 - Prob. 7PSCh. 9 - Prob. 8PSCh. 9 - Prob. 9PSCh. 9 - Prob. 10PSCh. 9 - Prob. 11PSCh. 9 - Prob. 12PSCh. 9 - Prob. 13PSCh. 9 - Prob. 14PSCh. 9 - Prob. 15PSCh. 9 - Prob. 16PS
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