Find the first seven terms of the sequence $$ a_{n}=2a_{n-2}+3a_{n-1}\text{ with a}_1 $$$$ =2\text{ and a}_2=5 $$

To find the first seven terms of the sequence defined by the recurrence relation

\[
a_{n} = 2a_{n-2} + 3a_{n-1}
\]

with initial conditions $a_1 = 2$ and $a_2 = 5$, we will calculate each term step by step.

### Step 1: Calculate $a_3$
Using the relation for $n = 3$:
\[
a_3 = 2a_1 + 3a_2 = 2(2) + 3(5) = 4 + 15 = 19
\]

### Step 2: Calculate $a_4$
Using the relation for $n = 4$:
\[
a_4 = 2a_2 + 3a_3 = 2(5) + 3(19) = 10 + 57 = 67
\]

### Step 3: Calculate $a_5$
Using the relation for $n = 5$:
\[
a_5 = 2a_3 + 3a_4 = 2(19) + 3(67) = 38 + 201 = 239
\]

### Step 4: Calculate $a_6$
Using the relation for $n = 6$:
\[
a_6 = 2a_4 + 3a_5 = 2(67) + 3(239) = 134 + 717 = 851
\]

### Step 5: Calculate $a_7$
Using the relation for $n = 7$:
\[
a_7 = 2a_5 + 3a_6 = 2(239) + 3(851) = 478 + 2553 = 3031
\]

### First Seven Terms
The first seven terms of the sequence are:
\[
a_1 = 2, \, a_2 = 5, \, a_3 = 19, \, a_4 = 67, \, a_5 = 239, \, a_6 = 851, \, a_7 = 3031
\]

Would you like to go over the details of the calculations or have any questions?

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Here are 5 related questions for further exploration:
1. How would the sequence change if $a_1 = 3$ and $a_2 = 4$?
2. What is the general form of the sequence $a_n$ without initial values?
3. Can you derive a closed-form expression for this sequence?
4. How does the sequence behave as $n$ becomes large?
5. How would the sequence be affected if the recurrence relation included a constant term?

**Tip:** When working with recurrence relations, it's helpful to calculate a few terms by hand to identify any patterns that might simplify the process of finding further terms.

Explore how to calculate the first seven terms of a sequence defined by the recurrence relation $a_{n} = 2a_{n-2} + 3a_{n-1}$ with initial conditions $a_1 = 2$ and $a_2 = 5$. Understand the step-by-step process and the mathematical concepts of recurrence relations and sequences. Suitable for high school students.

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