To solve this problem, we'll first use the general formula for the $n$th term of an arithmetic sequence:

$a_n = a_1 + (n - 1)d$

where:

• $a_n$ is the $n$th term,
• $a_1$ is the first term,
• $d$ is the common difference.

Step 1: Set up equations based on the given terms

Given:

• The third term $a_3 = 7$,
• The sixth term $a_6 = 16$.

We can write two equations using the general formula:

1. For the third term: $a_3 = a_1 + 2d = 7$

2. For the sixth term: $a_6 = a_1 + 5d = 16$

Step 2: Subtract the first equation from the second

Now, subtract the equation for $a_3$ from the equation for $a_6$:

$(a_1 + 5d) - (a_1 + 2d) = 16 - 7$

Simplifying this, we get:

$3d = 9$

So,

$d = 3$

Step 3: Find the first term $a_1$

Now that we have $d = 3$, we can substitute it back into the equation for $a_3$ to find $a_1$:

$a_1 + 2(3) = 7$

$a_1 + 6 = 7$

$a_1 = 1$

Step 4: Calculate the ninth term $a_9$

Finally, using the first term and the common difference, we can find the ninth term $a_9$:

$a_9 = a_1 + 8d = 1 + 8(3) = 1 + 24 = 25$

Final Answer

• The common difference $d$ is 3.
• The ninth term $a_9$ is 25.

Would you like further details or have any questions?

Here are some related questions:

1. How would you find the sum of the first 10 terms of this arithmetic sequence?
2. What is the 15th term of this arithmetic sequence?
3. How would the value of $d$ change if the third term was negative?
4. What happens to the sequence if the common difference $d$ is zero?
5. How would you determine the first term if only the common difference and a later term are known?

Tip: In arithmetic sequences, the difference between any two consecutive terms remains constant. This is key to setting up equations based on different terms in the sequence.