What is the value of `1^2+3^2+5^2+……15^2`?

Options

A. A)680
B. B)690
C. C)710
D. D)720

Answer

To find the value of \(1^2 + 3^2 + 5^2 + \ldots + 15^2\), we can follow these steps:

### Step 1: Identify the series
The series consists of the squares of all odd numbers from 1 to 15. The odd numbers in this range are:
1, 3, 5, 7, 9, 11, 13, 15.

### Step 2: Write the series in summation form
We can express the series as:
\[ S = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 + 11^2 + 13^2 + 15^2 \]

### Step 3: Calculate the squares of the odd numbers
Now, we will calculate the squares of each of these odd numbers:
- \(1^2 = 1\)
- \(3^2 = 9\)
- \(5^2 = 25\)
- \(7^2 = 49\)
- \(9^2 = 81\)
- \(11^2 = 121\)
- \(13^2 = 169\)
- \(15^2 = 225\)

### Step 4: Sum the squares
Now, we will sum these squares:
\[ S = 1 + 9 + 25 + 49 + 81 + 121 + 169 + 225 \]

Calculating this step by step:
- \(1 + 9 = 10\)
- \(10 + 25 = 35\)
- \(35 + 49 = 84\)
- \(84 + 81 = 165\)
- \(165 + 121 = 286\)
- \(286 + 169 = 455\)
- \(455 + 225 = 680\)

Thus, the total sum \(S\) is:
\[ S = 680 \]

### Final Answer
The value of \(1^2 + 3^2 + 5^2 + \ldots + 15^2\) is \(680\).

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