Suite numérique 
1)-Placer 6 nombres entre 39 et 59 de façon à obtenir une suite Arithmétique de 8 termes 
2)-Determiner une Suite Arithmétique de 10 termes dont la base est 10 et la somme 325
3)- Dans imune Suite Arithmétique dont la base est égal à la raison ,le 200éme terme est égal à 80. Quelle est la base et la raison ?
4)- Determiner une Suite  Arithmétique de 7 termes dont la base est 11 et la somme 203.
5)-Une Suite Arithmétique de 17 terme à pour terme central 45 et pour raison 8. Écrire tous les termes de cette Suite.
6)- Déterminer la raison de  cette Suite Géométrique si son 5 ém terme est 8 et son 8 ém terme vaut 67.

Let's solve each of these problems step by step.

### 1) Placer 6 nombres entre 39 et 59 de façon à obtenir une suite arithmétique de 8 termes

In a suite arithmétique (arithmetic sequence), the difference between consecutive terms is constant. We need to place 6 numbers between 39 and 59, forming an arithmetic sequence with 8 terms total.

Let the first term be \(a_1 = 39\) and the last term be \(a_8 = 59\). The common difference \(d\) can be determined by:

\[
a_n = a_1 + (n-1) \cdot d
\]

For \(a_8 = 59\):

\[
59 = 39 + 7d
\]

\[
20 = 7d
\]

\[
d = \frac{20}{7} \approx 2.857
\]

So, the 8-term arithmetic sequence is:

\[
39, 41.857, 44.714, 47.571, 50.429, 53.286, 56.143, 59
\]

The 6 numbers between 39 and 59 are:

\[
41.857, 44.714, 47.571, 50.429, 53.286, 56.143
\]

### 2) Déterminer une Suite Arithmétique de 10 termes dont la base est 10 et la somme est 325

For a sequence of 10 terms with a base \(a_1 = 10\) and sum \(S_{10} = 325\), we use the sum formula:

\[
S_n = \frac{n}{2} \times \left(2a_1 + (n-1)d\right)
\]

Substitute the known values:

\[
325 = \frac{10}{2} \times \left(2 \times 10 + 9d\right)
\]

\[
325 = 5 \times (20 + 9d)
\]

\[
325 = 100 + 45d
\]

\[
225 = 45d
\]

\[
d = 5
\]

So, the sequence is \(10, 15, 20, 25, 30, 35, 40, 45, 50, 55\).

### 3) Dans une Suite Arithmétique dont la base est égale à la raison, le 200ème terme est égal à 80. Quelle est la base et la raison ?

Given that the base \(a_1\) is equal to the common difference \(d\), and the 200th term \(a_{200} = 80\):

\[
a_n = a_1 + (n-1) \cdot d
\]

\[
80 = a_1 + 199a_1
\]

\[
80 = 200a_1
\]

\[
a_1 = \frac{80}{200} = 0.4
\]

So, the base and the common difference are both \(0.4\).

### 4) Déterminer une Suite Arithmétique de 7 termes dont la base est 11 et la somme est 203

We have \(a_1 = 11\), \(S_7 = 203\):

\[
S_n = \frac{n}{2} \times \left(2a_1 + (n-1)d\right)
\]

\[
203 = \frac{7}{2} \times (22 + 6d)
\]

\[
406 = 7 \times (22 + 6d)
\]

\[
406 = 154 + 42d
\]

\[
252 = 42d
\]

\[
d = 6
\]

So, the sequence is \(11, 17, 23, 29, 35, 41, 47\).

### 5) Une Suite Arithmétique de 17 termes a pour terme central 45 et pour raison 8. Écrire tous les termes de cette Suite.

The central term \(a_9 = 45\), and the common difference \(d = 8\). The sequence can be constructed by starting at \(a_9\) and moving backwards and forwards:

\[
a_8 = 45 - 8 = 37, \quad a_7 = 37 - 8 = 29, \quad \ldots
\]

\[
a_1 = 45 - 8 \times 8 = -19
\]

So, the sequence is:

\[
-19, -11, -3, 5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93, 101, 109
\]

### 6) Déterminer la raison de cette Suite Géométrique si son 5ème terme est 8 et son 8ème terme vaut 67

For a geometric sequence, the \(n\)th term is given by:

\[
a_n = a_1 \cdot r^{n-1}
\]

Let \(a_5 = 8\) and \(a_8 = 67\):

\[
\frac{a_8}{a_5} = \frac{a_1 \cdot r^7}{a_1 \cdot r^4} = r^3
\]

\[
r^3 = \frac{67}{8}
\]

\[
r = \sqrt[3]{\frac{67}{8}} \approx 2.08
\]

So, the common ratio \(r \approx 2.08\).

---

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### Related Questions:
1. How do you find the sum of an arithmetic series with a given number of terms?
2. What is the formula for the sum of a geometric series?
3. How can you determine the number of terms in an arithmetic sequence if you know the first term, the common difference, and the sum?
4. How does the common difference affect the rate of increase or decrease in an arithmetic sequence?
5. What are the properties of geometric sequences and how do they differ from arithmetic sequences?

### Tip:
In solving problems involving sequences, carefully differentiate between arithmetic and geometric sequences, and always use the appropriate formula for each type.

Learn how to solve various arithmetic and geometric sequence problems with detailed step-by-step solutions. Topics include determining sequence terms, finding common differences or ratios, and calculating sequence sums. Suitable for high school students.

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