Let's break down the problem step by step:

You have an arithmetic sequence where the first term is $x_1 = 5$, and the sum of the 4th and 5th terms is given as $x_4 + x_5 = 24$. You are asked to find the 7th term, $x_7$.

Step 1: General formula for terms of an arithmetic sequence

In an arithmetic sequence, the $n$-th term is given by the formula: $x_n = x_1 + (n-1) \cdot d$ where:

• $x_1$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Step 2: Expression for $x_4$ and $x_5$

• For the 4th term: $x_4 = x_1 + (4-1) \cdot d = 5 + 3d$
• For the 5th term: $x_5 = x_1 + (5-1) \cdot d = 5 + 4d$

Step 3: Use the condition $x_4 + x_5 = 24$

We are given that: $x_4 + x_5 = 24$ Substitute the expressions for $x_4$ and $x_5$: $(5 + 3d) + (5 + 4d) = 24$ Simplify: $10 + 7d = 24$ Solve for $d$: $7d = 14 \quad \Rightarrow \quad d = 2$

Step 4: Find $x_7$

Now that we know the common difference $d = 2$, we can find the 7th term $x_7$ using the formula for the $n$-th term: $x_7 = x_1 + (7-1) \cdot d = 5 + 6 \cdot 2 = 5 + 12 = 17$

Final Answer:

The 7th term of the arithmetic sequence is $x_7 = 17$.

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Would you like further details or have any questions on this?

Here are 5 related questions to explore:

1. How would you calculate the sum of the first 7 terms of an arithmetic sequence?
2. What happens if the first term of the arithmetic sequence was different (e.g., $x_1 = 3$)?
3. How can you find the $n$-th term of an arithmetic sequence if you are given the sum of the first $n$ terms?
4. What is the relationship between the common difference and the sum of consecutive terms in an arithmetic sequence?
5. Can the common difference be negative, and how would that affect the terms?

Tip: If you're ever unsure about the common difference in an arithmetic sequence, remember it can be found by subtracting any term from the next one!