Let's solve the problems one by one.

Problem 62

Problem: The arithmetic mean between two terms in an arithmetic sequence is 39. If one of these terms is 32, find the other term.

Solution: Let the two terms be $a$ and $b$. The arithmetic mean between them is given by: $\text{Arithmetic Mean} = \frac{a + b}{2}$ Given that the arithmetic mean is 39, we have: $\frac{a + b}{2} = 39$ This simplifies to: $a + b = 78$ Given $a = 32$, we find $b$ as follows: $32 + b = 78 \implies b = 78 - 32 = 46$ Answer: The other term is 46.

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Problem 63

Problem: If five arithmetic means are inserted between $-9$ and $9$, what is the third mean?

Solution: Let the six terms be $a_1, a_2, a_3, a_4, a_5, a_6, a_7$, where $a_1 = -9$ and $a_7 = 9$.

The common difference $d$ is given by: $d = \frac{a_7 - a_1}{6} = \frac{9 - (-9)}{6} = \frac{18}{6} = 3$

The third mean is $a_3$, and it is calculated as: $a_3 = a_1 + 2d = -9 + 2(3) = -9 + 6 = -3$ Answer: The third mean is $-3$.

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Problem 64

Problem: What is the first term of an arithmetic sequence when its arithmetic means are 35, 15, and $-5$?

Solution: Let's assume the arithmetic sequence has terms $a_1, a_2, a_3, a_4, a_5, a_6$, where the given arithmetic means correspond to $a_2 = 35$, $a_3 = 15$, and $a_4 = -5$.

The common difference $d$ between the terms can be found using: $d = a_3 - a_2 = 15 - 35 = -20$

The first term $a_1$ can be calculated as: $a_1 = a_2 - d = 35 - (-20) = 35 + 20 = 55$ Answer: The first term is 55.

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Problem 65

Problem: What is the last term of an arithmetic sequence when its arithmetic means are 35, 15, and $-5$?

Solution: Continuing from the previous problem, we found the common difference $d = -20$ and the first term $a_1 = 55$.

Now, we calculate the last term $a_6$ of the sequence: $a_6 = a_1 + 5d = 55 + 5(-20) = 55 - 100 = -45$ Answer: The last term is $-45$.

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Do you want any details or have any questions? Here are some related questions:

1. How do you determine the common difference in an arithmetic sequence?
2. What is the formula to find the nth term of an arithmetic sequence?
3. How do you find the sum of an arithmetic series?
4. Can you derive the formula for the arithmetic mean between two numbers?
5. How would the problem change if a geometric sequence were involved instead of an arithmetic sequence?

Tip: Always verify the common difference in arithmetic sequences by calculating consecutive terms to ensure consistency.