Unit 1 Sequences and Functions Assessment

Which of the following sequences is an arithmetic sequence?

• 2, 4, 8, 16, ...

• 5, 10, 15, 20, ...

• 3, 9, 27, 81, ...

• 1, 1, 2, 3, 5, ...

Answer explanation

The sequence 5, 10, 15, 20, ... is arithmetic because it has a constant difference of 5 between consecutive terms. The other sequences either double or follow a different pattern, not a constant addition.

Write the explicit formula for the arithmetic sequence: 10, 15, 20, 25, ...

Answer explanation

The first term is 10 and the common difference is 5. The explicit formula for an arithmetic sequence is given by a_n = a_1 + (n-1)d. Here, a_1 = 10 and d = 5, leading to a_n = 10 + 5(n-1) = 5n + 10.

What is the common difference in the arithmetic sequence: 7, 11, 15, 19, ...?

• 2

• 3

• 4

• 5

Answer explanation

In the arithmetic sequence 7, 11, 15, 19, the common difference is found by subtracting consecutive terms. 11 - 7 = 4, 15 - 11 = 4, and 19 - 15 = 4. Thus, the common difference is 4.

Which of the following sequences is a geometric sequence?

• 1, 2, 3, 4, ...

• 2, 4, 8, 16, ...

• 5, 10, 15, 20, ...

• 3, 6, 9, 12, ...

Answer explanation

The sequence 2, 4, 8, 16, ... is geometric because each term is multiplied by 2 to get the next term. The other sequences are arithmetic, where the difference between terms is constant.

What is the common ratio in the geometric sequence: 3, 9, 27, 81, ...?

• 2

• 3

• 4

• 5

Answer explanation

In a geometric sequence, the common ratio is found by dividing any term by the previous term. Here, 9/3 = 3, 27/9 = 3, and 81/27 = 3. Thus, the common ratio is 3.

Write the explicit formula for the geometric sequence: 5, 10, 20, 40, ...

Answer explanation

The first term is 5 and the common ratio is 2 (10/5 = 2). The explicit formula for a geometric sequence is given by a_n = a_1 * r^(n-1). Thus, a_n = 5 * 2^(n-1) is the correct choice.

• 7

• 10

• 13

• 16

Answer explanation

The sequence starts with a_1 = 4 and each term increases by 3. Thus, a_2 = 7, a_3 = 10, a_4 = 13, and a_5 = 16. Therefore, the 5th term is 16.

• 4

• 8

• 16

• 32

Answer explanation

The sequence starts with a_1 = 2. Each term is double the previous one: a_2 = 4, a_3 = 8, and a_4 = 16. Thus, the 4th term is 16.

Write the recursive formula for the sequence: 12, 24, 48, 96, ...

Answer explanation

The sequence doubles each term: 12, 24 (12*2), 48 (24*2), 96 (48*2). Thus, the recursive formula is a_1 = 12, a_n = 2 * a_{n-1}, which correctly describes the pattern of the sequence.

Answer explanation

The population doubles every month, indicating exponential growth. The correct model is y = 50 * 2^x, where 50 is the initial population and 2^x represents the doubling effect over x months.