The tenth tern of an A.P. `sqrt(2),3sqrt(2),5sqrt(2),7sqrt(2)` . . . . . Is :

Options

A. `11sqrt(2)`
B. 12
C. `10sqrt(2)`
D. `19sqrt(2)`

Answer

To find the 10th term of the arithmetic progression (A.P.) given as \( \sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, \ldots \), we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the A.P. is:
\[ a = \sqrt{2} \]

Next, we need to find the common difference \( d \). The common difference can be calculated by subtracting the first term from the second term:
\[ d = 3\sqrt{2} - \sqrt{2} = 2\sqrt{2} \]

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term \( A_n \) of an A.P. is given by:
\[ A_n = a + (n - 1) \cdot d \]

### Step 3: Substitute the values into the formula
We want to find the 10th term, so we set \( n = 10 \):
\[ A_{10} = \sqrt{2} + (10 - 1) \cdot (2\sqrt{2}) \]

### Step 4: Simplify the expression
Now we can simplify the expression:
\[ A_{10} = \sqrt{2} + 9 \cdot (2\sqrt{2}) \]
\[ A_{10} = \sqrt{2} + 18\sqrt{2} \]
\[ A_{10} = 19\sqrt{2} \]

### Final Answer
Thus, the 10th term of the A.P. is:
\[ \boxed{19\sqrt{2}} \]

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