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Problem 66

Problem: Find the value of $x$ if the arithmetic mean of 3 and $(3x + 5)$ is 8.

Solution: The arithmetic mean of two numbers $a$ and $b$ is given by: $\text{Arithmetic Mean} = \frac{a + b}{2}$ Here, $a = 3$ and $b = 3x + 5$. We are given that the arithmetic mean is 8, so: $\frac{3 + (3x + 5)}{2} = 8$ Simplify and solve for $x$: $\frac{3 + 3x + 5}{2} = 8$ $\frac{3x + 8}{2} = 8$ Multiply both sides by 2: $3x + 8 = 16$ Subtract 8 from both sides: $3x = 8$ Finally, divide by 3: $x = \frac{8}{3}$ Answer: The value of $x$ is $\frac{8}{3}$.

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Problem 67

Problem: Find the value of $a$ when the arithmetic mean of $(a+7)$ and $(a+3)$ is $3a + 9$.

Solution: The arithmetic mean of $a+7$ and $a+3$ is given by: $\frac{(a + 7) + (a + 3)}{2} = 3a + 9$ Simplify the left-hand side: $\frac{a + 7 + a + 3}{2} = 3a + 9$ $\frac{2a + 10}{2} = 3a + 9$ Simplify further: $a + 5 = 3a + 9$ Subtract $a$ from both sides: $5 = 2a + 9$ Subtract 9 from both sides: $-4 = 2a$ Finally, divide by 2: $a = -2$ Answer: The value of $a$ is $-2$.

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Would you like more details or have any questions? Here are some related questions:

1. What steps should be followed when solving for variables in arithmetic mean problems?
2. How can arithmetic means be used to form equations?
3. What other types of sequences are commonly used in algebra, aside from arithmetic sequences?
4. How do you solve a problem where the mean of three or more terms is given?
5. What is the geometric mean, and how does it differ from the arithmetic mean?

Tip: When dealing with arithmetic mean problems, carefully simplify and arrange the terms before isolating the variable to solve the equation.