Find the sum of integers between 120 and 365 which are divisible by 9.

To find the sum of integers between 120 and 365 that are divisible by 9, follow these steps:

### Step 1: Identify the first and last terms
1. **First term**: The smallest integer divisible by 9 greater than or equal to 120 is \( 126 \) (since \( 9 \times 14 = 126 \)).
2. **Last term**: The largest integer divisible by 9 less than or equal to 365 is \( 360 \) (since \( 9 \times 40 = 360 \)).

So, we are working with an arithmetic sequence that starts at \( a = 126 \) and ends at \( l = 360 \) with a common difference \( d = 9 \).

### Step 2: Find the number of terms in the sequence
To find the number of terms \( n \), use the formula for the \( n \)-th term of an arithmetic sequence:

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

Set \( a_n = 360 \) and solve for \( n \):

\[
360 = 126 + (n - 1) \times 9
\]
\[
234 = (n - 1) \times 9
\]
\[
n - 1 = \frac{234}{9} = 26
\]
\[
n = 27
\]

So, there are \( 27 \) terms in this sequence.

### Step 3: Calculate the sum of the sequence
The sum \( S_n \) of an arithmetic sequence can be calculated with the formula:

\[
S_n = \frac{n}{2} \times (a + l)
\]

Substitute \( n = 27 \), \( a = 126 \), and \( l = 360 \):

\[
S_{27} = \frac{27}{2} \times (126 + 360)
\]
\[
S_{27} = \frac{27}{2} \times 486
\]
\[
S_{27} = 27 \times 243 = 6561
\]

### Answer
The sum of integers between 120 and 365 that are divisible by 9 is **6561**.

Would you like further details or have any questions?

Here are some related questions to deepen understanding:
1. What is the general approach to find the sum of numbers in a sequence with any common divisor?
2. How would the approach differ if you needed numbers divisible by both 9 and another number?
3. Can you generalize this process to find sums within any arbitrary range?
4. What adjustments are needed if the sequence had non-integer terms or a fractional divisor?
5. How does the sum formula change if the sequence is not arithmetic but geometric?

**Tip:** For arithmetic sequences, verify both the first and last terms meet the conditions to avoid extra calculations.

This guide helps you find the sum of integers between 120 and 365 that are divisible by 9. Using arithmetic sequence formulas, we determine the sum is 6561. Suitable for students in Grades 9-11 learning about sequences and summations.

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