A is a series of five consecutive even numbers and B is a series of five consecutive odd numbers. The third term of series A is 3 more than the third term of series B. If the sum of the first terms of both series is 397, what is the sum of the third terms of series A and B?

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Accepted Answer

**Solution:**

1. Let the five consecutive even numbers in series A be: $a, a+2, a+4, a+6, a+8$.

2. Let the five consecutive odd numbers in series B be: $b, b+2, b+4, b+6, b+8$.

3. According to the question, the third term of series A is 3 more than the third term of series B:
   $$a + 4 = b + 4 + 3$$
   Simplifying, we get:
   $$a - b = 3 	ag{i}$$

4. The sum of the first terms of both series is 397:
   $$a + b = 397 	ag{ii}$$

5. Adding equations (i) and (ii):
   $$a - b + a + b = 3 + 397$$
   $$2a = 400 ightarrow a = 200$$

6. Substitute $a = 200$ into equation (ii):
   $$200 + b = 397 ightarrow b = 197$$

7. The third term of series A is:
   $$a + 4 = 200 + 4 = 204$$

8. The third term of series B is:
   $$b + 4 = 197 + 4 = 201$$

9. Required sum of the third terms:
   $$204 + 201 = 405$$

**Answer:** 405

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