Exercise 124 Page 241

Let first talk about the difference between a linear and an exponential sequence. Note that an exponential sequence is the same thing as a geometric sequence.

If a sequence is linear, there is a common difference between consecutive terms.
If a sequence is geometric, there is a common ratio between consecutive terms.

Let's check if it is linear.

As we can see, we have a common difference. Therefore, this sequence is arithmetic.

Again, we will begin by examining if the sequence is arithmetic.

It is not arithmetic. Let's check if it is geometric.

There is a common ratio between consecutive terms. Therefore this is a geometric sequence.

Let's take a look at the given sequence.

Recall that an arithmetic sequence increases by a common difference, decreases by a common difference, or is constant. Increasing:& 1, 3, 5, 7,... Decreasing:& 4, 1, -2, -5,... Constant:& 4, 1, -2, -5,... On the other hand, a geometric sequence increases by a common factor, decreases by a common factor, changes sign for each term, or is constant. Increasing:& 1, 2, 4, 9,... Decreasing:& 32, 8, 2, 0.5,... Changing sign:& 1, -1, 1, -1 ... Constant:& 2, 2, 2, 2,... Examining the sequence, however, we see that it first decreases from 1 to 0 and then it increases from 0 to 1. Also, the next term is not the result of the previous term multiplied by the constant factor. Therefore, it is neither geometric nor arithmetic.

As in Parts A and B, we will first check if it is arithmetic.

There is a common difference, and therefore this sequence is arithmetic.

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Exercise 124 Page 241

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