What are the first three terms of the sequence whose 4th, 5th, and 6th terms are 3,2, and 6 respectively.



Let's denote the sequence by \( a_1, a_2, a_3, \ldots \). We know the following:

- \( a_4 = 3 \)
- \( a_5 = 2 \)
- \( a_6 = 6 \)

To find the first three terms, we need to establish a pattern or a formula that describes the sequence.

One approach is to assume that the sequence is arithmetic or geometric, but since neither an arithmetic nor a geometric sequence fits these terms directly, we should explore the possibility of a recurrence relation or another type of sequence.

Given the sequence values we have:

- \( a_4 = 3 \)
- \( a_5 = 2 \)
- \( a_6 = 6 \)

Let's hypothesize that the sequence follows a recurrence relation of the form:
\[
a_{n} = p \cdot a_{n-1} + q \cdot a_{n-2} + r \cdot a_{n-3}
\]
where \( p \), \( q \), and \( r \) are constants.

We substitute the known values to set up a system of equations:

1. For \( n = 6 \):
   \[
   a_6 = p \cdot a_5 + q \cdot a_4 + r \cdot a_3 
   \]
   Substituting the known values:
   \[
   6 = p \cdot 2 + q \cdot 3 + r \cdot a_3 
   \]
   
2. For \( n = 5 \):
   \[
   a_5 = p \cdot a_4 + q \cdot a_3 + r \cdot a_2 
   \]
   Substituting the known values:
   \[
   2 = p \cdot 3 + q \cdot a_3 + r \cdot a_2 
   \]

3. For \( n = 4 \):
   \[
   a_4 = p \cdot a_3 + q \cdot a_2 + r \cdot a_1 
   \]
   Substituting the known values:
   \[
   3 = p \cdot a_3 + q \cdot a_2 + r \cdot a_1 
   \]

This system of equations is challenging to solve without additional information, so another way is to assume the sequence follows a known pattern or generate a difference table to check for regularity in the sequence.

Given the lack of specific pattern rules, let's check if any simple arithmetic or geometric sequence fits:

### Arithmetic Check:
The difference between the terms does not suggest an arithmetic sequence since the differences aren't constant.

### Geometric Check:
The ratio between the terms doesn't suggest a geometric sequence since the ratio isn't constant.

### Possible Simple Relation:
Considering simple linear relations, let's test a Fibonacci-like sequence where each term could be a linear combination of the previous terms.

Testing with \( a_n = a_{n-2} + a_{n-3} \) gives us:
\[
a_6 = a_4 + a_3 \quad \text{or} \quad 6 = 3 + a_3 \quad \Rightarrow \quad a_3 = 3
\]
\[
a_5 = a_3 + a_2 \quad \text{or} \quad 2 = 3 + a_2 \quad \Rightarrow \quad a_2 = -1
\]
\[
a_4 = a_2 + a_1 \quad \text{or} \quad 3 = -1 + a_1 \quad \Rightarrow \quad a_1 = 4
\]

So the first three terms of the sequence are \( a_1 = 4 \), \( a_2 = -1 \), and \( a_3 = 3 \).

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

1. How can you determine the general formula for a sequence given certain terms?
2. What are recurrence relations, and how do they work?
3. What are the differences between arithmetic and geometric sequences?
4. How can difference tables be used to find patterns in sequences?
5. What are some common sequences used in mathematical problems?

**Tip:** When dealing with sequences, always check if they can be expressed as an arithmetic or geometric sequence before moving on to more complex relations.

Learn how to find the first three terms of a sequence when the 4th, 5th, and 6th terms are given. This problem involves sequences and recurrence relations, exploring different patterns to determine the initial terms. Suitable for students in Grades 10-12.

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