Observe the following pattern and find the sum of `1 + 3 + 5 + 7 + 9 + ... + 19. `
`{:(1" " = 1 xx 1 = 1 ),(1 + 3 " " = 2 xx 2 = 4 ),(1 + 3 + 5 " " = 3 xx 3 = 9 ),(1 + 3 + 5 + 7 " " = 4 xx 4 = 16 ),(1 + 3 + 5 + 7 + 9 " "= 5 xx 5 = 25 ):}`

Options

A. 121
B. 81
C. 100
D. 64

Answer

To find the sum of the series \(1 + 3 + 5 + 7 + 9 + ... + 19\), we can observe the pattern in the series of odd numbers. Let's break it down step by step.

### Step 1: Identify the Pattern
The series consists of the first \(n\) odd numbers. The odd numbers start from 1 and continue as follows:
- 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.

### Step 2: Count the Numbers
Next, we need to count how many odd numbers are there from 1 to 19.
- The odd numbers are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.
- There are a total of 10 odd numbers.

### Step 3: Use the Formula for the Sum of Odd Numbers
The sum of the first \(n\) odd numbers can be calculated using the formula:
\[ \text{Sum} = n \times n \]
where \(n\) is the number of terms.

### Step 4: Apply the Formula
Since we have identified that there are 10 odd numbers:
\[ \text{Sum} = 10 \times 10 = 100 \]

### Conclusion
Thus, the sum of the series \(1 + 3 + 5 + 7 + 9 + ... + 19\) is \(100\).

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