"Patterns & Algebra" math worksheet featuring multiple-choice questions on number sequences, equations, and problem-solving.

Let's solve each problem step by step.

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Problem 1: What is the missing number from the pattern?

Pattern: 5, 11, 23, __, 95, 191, 383

#### Solution:
- Observe the differences between consecutive terms:
- \( 11 - 5 = 6 \)
- \( 23 - 11 = 12 \)
- The differences are increasing by a factor of 2.
- Next difference should be \( 12 \times 2 = 24 \).
- Therefore, the next term is \( 23 + 24 = 47 \).

Answer: \( \boxed{47} \)

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Problem 2: Enter the missing number to make the number sentence true.

Equation: \( 6^2 \times 8 = \_\_ - 5^2 \)

#### Solution:
- Calculate \( 6^2 \):
\[
6^2 = 36
\]
- Multiply by 8:
\[
36 \times 8 = 288
\]
- Calculate \( 5^2 \):
\[
5^2 = 25
\]
- Set up the equation:
\[
288 = x - 25
\]
- Solve for \( x \):
\[
x = 288 + 25 = 313
\]

Answer: \( \boxed{313} \)

---

Problem 3:

Given:
\[
a = 11, \quad b = 9, \quad c = 12
\]
Find:
\[
\frac{a + b \times c}{2}
\]

#### Solution:
- First, calculate \( b \times c \):
\[
b \times c = 9 \times 12 = 108
\]
- Add \( a \) to \( b \times c \):
\[
a + b \times c = 11 + 108 = 119
\]
- Divide by 2:
\[
\frac{119}{2} = 59.5
\]

Answer: \( \boxed{59.5} \)

---

Problem 4:

Given:
\[
a + b = 19, \quad a \times c = 96, \quad c - b = 1
\]
\( a \), \( b \), and \( c \) are whole numbers. Find the value of \( a \).

#### Solution:
1. From \( a + b = 19 \), express \( b \) in terms of \( a \):
\[
b = 19 - a
\]
2. From \( c - b = 1 \), express \( c \) in terms of \( b \):
\[
c = b + 1 = (19 - a) + 1 = 20 - a
\]
3. Substitute \( c = 20 - a \) into \( a \times c = 96 \):
\[
a \times (20 - a) = 96
\]
\[
20a - a^2 = 96
\]
\[
a^2 - 20a + 96 = 0
\]
4. Solve the quadratic equation using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[
a = \frac{20 \pm \sqrt{20^2 - 4 \cdot 1 \cdot 96}}{2 \cdot 1}
\]
\[
a = \frac{20 \pm \sqrt{400 - 384}}{2}
\]
\[
a = \frac{20 \pm \sqrt{16}}{2}
\]
\[
a = \frac{20 \pm 4}{2}
\]
\[
a = \frac{24}{2} = 12 \quad \text{or} \quad a = \frac{16}{2} = 8
\]
5. Check both solutions:
- If \( a = 12 \):
\[
b = 19 - 12 = 7, \quad c = 20 - 12 = 8
\]
\[
a \times c = 12 \times 8 = 96 \quad (\text{correct})
\]
- If \( a = 8 \):
\[
b = 19 - 8 = 11, \quad c = 20 - 8 = 12
\]
\[
a \times c = 8 \times 12 = 96 \quad (\text{correct})
\]
Both solutions work, but typically the smaller value is preferred unless specified otherwise.

Answer: \( \boxed{8} \)

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Problem 5: Find the sum of the numbers from 1 to 100.

#### Solution:
Use the formula for the sum of the first \( n \) natural numbers:
\[
S_n = \frac{n(n + 1)}{2}
\]
Here, \( n = 100 \):
\[
S_{100} = \frac{100(100 + 1)}{2} = \frac{100 \times 101}{2} = 5050
\]

Answer: \( \boxed{5050} \)

---

Problem 6:

Given:
\[
a \text{ is double } b, \quad b \text{ is a third of } c, \quad d \text{ is 4 times } a
\]
Which of the following is true?
- \( d = 8 \times b \)
- \( b = d \times 3 \)
- \( a = 2 \times c \)
- \( b = a \times c \)

#### Solution:
1. Express relationships:
- \( a = 2b \)
- \( b = \frac{c}{3} \Rightarrow c = 3b \)
- \( d = 4a \)
2. Substitute \( a = 2b \) into \( d = 4a \):
\[
d = 4(2b) = 8b
\]
This matches \( d = 8 \times b \).

Answer: \( \boxed{d = 8 \times b} \)

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Problem 7: What is the 25th number in this pattern?

Pattern: 0.75, 1.5, 2.25, 3, ...

#### Solution:
- The pattern increases by 0.75 each time.
- General term of an arithmetic sequence:
\[
a_n = a_1 + (n - 1)d
\]
Here, \( a_1 = 0.75 \), \( d = 0.75 \), and \( n = 25 \):
\[
a_{25} = 0.75 + (25 - 1) \times 0.75
\]
\[
a_{25} = 0.75 + 24 \times 0.75
\]
\[
a_{25} = 0.75 + 18 = 18.75
\]

Answer: \( \boxed{18.75} \)

---

Problem 8:

Given:
\[
a \times a \times a + a = 520, \quad b \text{ is double } a
\]
Which of the following is true?
- \( a \times a = b^2 \)
- \( a \times b = 126 \)
- \( b \times b = 256 \)
- \( a + b \times b = 384 \)

#### Solution:
1. Simplify the given equation:
\[
a^3 + a = 520
\]
Factor out \( a \):
\[
a(a^2 + 1) = 520
\]
2. Test integer values for \( a \):
- Try \( a = 8 \):
\[
8^3 + 8 = 512 + 8 = 520 \quad (\text{correct})
\]
- So, \( a = 8 \).
3. Since \( b \) is double \( a \):
\[
b = 2a = 2 \times 8 = 16
\]
4. Check each option:
- \( a \times a = b^2 \):
\[
a^2 = 8^2 = 64, \quad b^2 = 16^2 = 256 \quad (\text{false})
\]
- \( a \times b = 126 \):
\[
a \times b = 8 \times 16 = 128 \quad (\text{false})
\]
- \( b \times b = 256 \):
\[
b^2 = 16^2 = 256 \quad (\text{true})
\]
- \( a + b \times b = 384 \):
\[
a + b^2 = 8 + 256 = 264 \quad (\text{false})
\]

Answer: \( \boxed{b \times b = 256} \)

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Problem 9: Fill in the missing number.

Pattern: 225, 196, 169, __, 121

#### Solution:
- The numbers are perfect squares in descending order:
\[
225 = 15^2, \quad 196 = 14^2, \quad 169 = 13^2, \quad \_, \quad 121 = 11^2
\]
- The missing term is \( 12^2 \):
\[
12^2 = 144
\]

Answer: \( \boxed{144} \)

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Problem 10: What is the hundredth number in this pattern?

Pattern: 8, 14, 20, 26, 32, ...

#### Solution:
- The pattern increases by 6 each time.
- General term of an arithmetic sequence:
\[
a_n = a_1 + (n - 1)d
\]
Here, \( a_1 = 8 \), \( d = 6 \), and \( n = 100 \):
\[
a_{100} = 8 + (100 - 1) \times 6
\]
\[
a_{100} = 8 + 99 \times 6
\]
\[
a_{100} = 8 + 594 = 602
\]

Answer: \( \boxed{602} \)

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Final Answers:

1. \( \boxed{47} \)
2. \( \boxed{313} \)
3. \( \boxed{59.5} \)
4. \( \boxed{8} \)
5. \( \boxed{5050} \)
6. \( \boxed{d = 8 \times b} \)
7. \( \boxed{18.75} \)
8. \( \boxed{b \times b = 256} \)
9. \( \boxed{144} \)
10. \( \boxed{602} \)