What is the value of `9^(2) + 11 ^(2) +… 21 ^(2)` ?

Options

A. 1441
B. 1346
C. 1532
D. 1687

Answer

To find the value of \(9^2 + 11^2 + 13^2 + 15^2 + 17^2 + 19^2 + 21^2\), we can follow these steps:

### Step 1: Identify the Sequence
The numbers \(9, 11, 13, 15, 17, 19, 21\) form an arithmetic sequence where the first term \(a = 9\) and the common difference \(d = 2\).

### Step 2: Determine the Number of Terms
To find the number of terms \(n\) in the sequence, we can use the formula for the \(n\)-th term of an arithmetic sequence:
\[ a_n = a + (n-1)d \]
Setting \(a_n = 21\):
\[ 21 = 9 + (n-1) \cdot 2 \]
\[ 21 - 9 = (n-1) \cdot 2 \]
\[ 12 = (n-1) \cdot 2 \]
\[ n - 1 = 6 \quad \Rightarrow \quad n = 7 \]
So, there are 7 terms in the sequence.

### Step 3: Calculate the Sum of Squares
We need to calculate the sum of squares of these terms:
\[ S = 9^2 + 11^2 + 13^2 + 15^2 + 17^2 + 19^2 + 21^2 \]

Calculating each square:
- \(9^2 = 81\)
- \(11^2 = 121\)
- \(13^2 = 169\)
- \(15^2 = 225\)
- \(17^2 = 289\)
- \(19^2 = 361\)
- \(21^2 = 441\)

Now, sum these values:
\[ S = 81 + 121 + 169 + 225 + 289 + 361 + 441 \]

### Step 4: Perform the Addition
Adding these values step by step:
1. \(81 + 121 = 202\)
2. \(202 + 169 = 371\)
3. \(371 + 225 = 596\)
4. \(596 + 289 = 885\)
5. \(885 + 361 = 1246\)
6. \(1246 + 441 = 1687\)

Thus, the final sum is:
\[ S = 1687 \]

### Conclusion
The value of \(9^2 + 11^2 + 13^2 + 15^2 + 17^2 + 19^2 + 21^2\) is \(1687\).

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