Without adding find the sum:
`(1+ 3+5 +7 +9)`
`(1+3+5+7+9+11+13+15+17)`

Answer

To solve the question of finding the sums of the sequences of odd numbers without adding them directly, we can use the formula for the sum of the first \( n \) odd numbers, which is \( n^2 \).

### Step-by-Step Solution:

1. **Identify the first sequence:**
The first sequence is \( 1 + 3 + 5 + 7 + 9 \).
- Here, the odd numbers are \( 1, 3, 5, 7, \) and \( 9 \).

2. **Count the number of terms:**
- There are \( 5 \) terms in this sequence.

3. **Use the formula for the sum of the first \( n \) odd numbers:**
- The formula states that the sum of the first \( n \) odd numbers is \( n^2 \).
- For \( n = 5 \), the sum is \( 5^2 = 25 \).

4. **Identify the second sequence:**
The second sequence is \( 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 \).
- The odd numbers in this sequence are \( 1, 3, 5, 7, 9, 11, 13, 15, \) and \( 17 \).

5. **Count the number of terms:**
- There are \( 9 \) terms in this sequence.

6. **Use the formula for the sum of the first \( n \) odd numbers again:**
- For \( n = 9 \), the sum is \( 9^2 = 81 \).

### Final Answers:
- The sum of the first sequence \( (1 + 3 + 5 + 7 + 9) \) is \( 25 \).
- The sum of the second sequence \( (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17) \) is \( 81 \).