First Summative Test Q1 Interactive Worksheet - Edform - Free Printable

Let's solve each problem step by step:

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Problem 1:

What is the next number in the sequence 36, 30, 24, 18, …?

- The sequence is decreasing.
- The difference between consecutive terms is:
- \( 36 - 30 = 6 \)
- \( 30 - 24 = 6 \)
- \( 24 - 18 = 6 \)

So, the common difference is \(-6\).

- The next term is:
\[
18 - 6 = 12
\]

Answer: a. 12

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Problem 2:

What is the nth term of the sequence 11, 16, 21, 26, …?

- The sequence is increasing.
- The difference between consecutive terms is:
- \( 16 - 11 = 5 \)
- \( 21 - 16 = 5 \)
- \( 26 - 21 = 5 \)

So, the common difference is \(5\).

- The nth term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1)d
\]
where \(a_1\) is the first term and \(d\) is the common difference.

- Here, \(a_1 = 11\) and \(d = 5\):
\[
a_n = 11 + (n-1) \cdot 5
\]
Simplify:
\[
a_n = 11 + 5n - 5 = 5n + 6
\]

Answer: c. \(a_n = 5n + 6\)

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Problem 3:

What is the 8th term of the sequence \(a_n = 6n - 7\)?

- The formula for the nth term is given as \(a_n = 6n - 7\).
- To find the 8th term (\(a_8\)):
\[
a_8 = 6(8) - 7 = 48 - 7 = 41
\]

Answer: d. 41

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Problem 4:

If the terms of an arithmetic sequence are in increasing order, then the common difference is:

- For an arithmetic sequence to be in increasing order, the common difference must be positive.
- If the common difference were negative, the sequence would be decreasing.
- If the common difference were zero, the sequence would be constant.

Answer: a. positive

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Problem 5:

Which of these following expressions represents the fourth term of an arithmetic sequence?

- The nth term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1)d
\]
- For the fourth term (\(n = 4\)):
\[
a_4 = a_1 + (4-1)d = a_1 + 3d
\]

Answer: d. \(a_1 + 3d\)

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Problem 6:

Which of the following describes an arithmetic sequence?

- An arithmetic sequence has a constant difference between consecutive terms.
- Let's analyze each option:
- a. \(a_n = 6n^2\): This is a quadratic sequence, not arithmetic.
- b. \(a_n = 6n\): This is linear, and the difference between consecutive terms is constant (\(6\)). So, it is arithmetic.
- c. \(a_n = \frac{1}{6n}\): This is not arithmetic because the difference between terms is not constant.
- d. \(a_n = 6^n\): This is exponential, not arithmetic.

Answer: b. \(a_n = 6n\)

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Problem 7:

Which of these words has the same meaning as arithmetic mean?

- The arithmetic mean is the average of a set of numbers.
- Therefore, the correct synonym is "average."

Answer: c. average

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Problem 8:

A term that is between two terms of an arithmetic sequence is called an arithmetic _______.

- A term that is between two terms of an arithmetic sequence is called an arithmetic mean.

Answer: d. mean

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Problem 9:

How many arithmetic means may be inserted between two terms of an arithmetic sequence?

- There can be any number of arithmetic means inserted between two terms of an arithmetic sequence. For example, between \(a\) and \(b\), you can insert 1, 2, 3, or more terms.

Answer: c. any number

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Problem 10:

Which of the following values cannot be a common ratio of a geometric sequence?

- The common ratio of a geometric sequence can be any real number except zero. If the common ratio were zero, the sequence would become trivial (all terms after the first would be zero).

Answer: c. 0

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Problem 11:

Which of the following is a geometric sequence?

- A geometric sequence has a constant ratio between consecutive terms.
- Let's analyze each option:
- a. 3, 7, 11, 15, …: The differences are constant (\(4\)), so this is arithmetic, not geometric.
- b. 1/2, 1/3, 2/9, …: The ratios are not constant.
- c. 4, 4.5, 5, …: The differences are constant (\(0.5\)), so this is arithmetic, not geometric.
- d. 2, 3, 5, 8, …: The differences are not constant, and the ratios are not constant.

None of these are geometric sequences.

Answer: None of these (d)

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Problem 12:

What is the common ratio of the sequence whose first term and third terms are 2 and 32, respectively?

- Let the first term be \(a_1 = 2\) and the third term be \(a_3 = 32\).
- The nth term of a geometric sequence is given by:
\[
a_n = a_1 \cdot r^{n-1}
\]
- For the third term (\(n = 3\)):
\[
a_3 = a_1 \cdot r^{3-1} = 2 \cdot r^2
\]
Given \(a_3 = 32\):
\[
2 \cdot r^2 = 32
\]
Solve for \(r^2\):
\[
r^2 = \frac{32}{2} = 16
\]
Take the square root:
\[
r = \pm 4
\]

Answer: b. ±4

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Problem 13:

An arithmetic sequence has a common difference while a geometric sequence has a common _______.

- A geometric sequence has a common ratio.

Answer: c. ratio

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Problem 14:

Which of the following is an arithmetic sequence?

- Let's analyze each option:
- a. \(1/2, 1, 3/2, …\): The differences are constant (\(1/2\)), so this is arithmetic.
- b. \(1, -1, 1, -1, …\): The differences are not constant.
- c. \(1/2, 1/4, 1/8, …\): The ratios are constant (\(1/2\)), so this is geometric.
- d. \(-2, 4, -8, …\): The ratios are constant (\(-2\)), so this is geometric.

Answer: a. \(1/2, 1, 3/2, …\)

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Problem 15:

Which of these is a geometric sequence?

- Let's analyze each option:
- a. \(2, 5, 8, 11, …\): The differences are constant (\(3\)), so this is arithmetic.
- b. \(1, 0, 0, 0, …\): The ratios are not constant.
- c. \(-2, 4, -8, …\): The ratios are constant (\(-2\)), so this is geometric.
- d. \(-2, -4, -6, …\): The differences are constant (\(-2\)), so this is arithmetic.

Answer: c. \(-2, 4, -8, …\)

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Final Answers:

1. a. 12
2. c. \(a_n = 5n + 6\)
3. d. 41
4. a. positive
5. d. \(a_1 + 3d\)
6. b. \(a_n = 6n\)
7. c. average
8. d. mean
9. c. any number
10. c. 0
11. d. none of these
12. b. ±4
13. c. ratio
14. a. \(1/2, 1, 3/2, …\)
15. c. \(-2, 4, -8, …\)

\boxed{a, c, d, a, d, b, c, d, c, c, d, b, c, a, c}