Exercise 57 Page 206

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a The difference between the consecutive terms of an arithmetic sequence is constant. This means that if there is a common difference between the consecutive terms of the sequence, then the sequence is arithmetic. Let's find the difference between the first and second terms.

a_{2} - a_{1}

SubstituteII

$a_{2} = 3 x + 6$ , $a_{1} = x + 6$

3 x + 6 - (x + 6)

Distr

Distribute $- 1$

3 x + 6 - x - 6

SubTerms

Subtract terms

2 x

Let's do the same thing for the rest of the terms.

$a_{2} - a_{1}$	$3 x + 6 - (x + 6) = 2 x$
$a_{3} - a_{2}$	$5 x + 6 - (3 x + 6) = 2 x$
$a_{3} - a_{2}$	$7 x + 6 - (5 x + 6) = 2 x$

In each case, the difference is constant and equal to

2 x .

This means that we have a common difference and therefore the given sequence is arithmetic.

b As in Part A, recall that the difference between the consecutive terms of an arithmetic sequence is constant. Let's calculate the difference between the consecutive terms of the given sequence.

$a_{2} - a_{1}$	$3 x + 1 - (x + 1) = 2 x$
$a_{3} - a_{2}$	$9 x + 1 - (3 x + 1) = 6 x$
$a_{3} - a_{2}$	$27 x + 1 - (9 x + 1) = 18 x$

Unlike in Part A, the difference between consecutive terms is not constant. This means that the given sequence is not arithmetic.

Subchapter links

Communicate Your Answer p.199

Monitoring Progress p.200-203

Exercises p.204-206

Exercise 57 Page 206

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