Step 1: Understanding the Concept:
An "even multiple of 9" is essentially a multiple of \( 9 \times 2 = 18 \). We need to find the sum of multiples of 18 in the range (300, 500). : Key Formula or Approach:
1. \( n\text{-th term of AP}: a_n = a + (n-1)d \).
2. \( \text{Sum of AP}: S_n = \frac{n}{2}(a + l) \). Step 2: Detailed Explanation:
1. Identify the first term (a):
\( 300 / 18 \approx 16.66 \). So the first multiple after 300 is \( 18 \times 17 = 306 \).
2. Identify the last term (l):
\( 500 / 18 \approx 27.77 \). So the last multiple before 500 is \( 18 \times 27 = 486 \).
3. Find the number of terms (n):
\[ 486 = 306 + (n - 1)18 \]
\[ 180 = (n - 1)18 \implies 10 = n - 1 \implies n = 11 \]
4. Calculate the sum:
\[ S = \frac{11}{2}(306 + 486) = \frac{11}{2}(792) \]
\[ S = 11 \times 396 = 4356 \]. Step 3: Final Answer:
The sum is 4356.