If `a=sqrt8-sqrt7,b=sqrt7-sqrt6` and `c=sqrt6-sqrt5` then which answer is right?

Options

A. a)a > b > c
B. b)a < b < c
C. c)b>a >c
D. d)a > c >b

Answer

To solve the problem, we need to compare the values of \( a \), \( b \), and \( c \) defined as follows:

- \( a = \sqrt{8} - \sqrt{7} \)
- \( b = \sqrt{7} - \sqrt{6} \)
- \( c = \sqrt{6} - \sqrt{5} \)

### Step 1: Compare \( a \) and \( b \)

We start by calculating \( a - b \):

\[ a - b = (\sqrt{8} - \sqrt{7}) - (\sqrt{7} - \sqrt{6}) \]
\[ = \sqrt{8} - \sqrt{7} - \sqrt{7} + \sqrt{6} \]
\[ = \sqrt{8} + \sqrt{6} - 2\sqrt{7} \]

To determine the sign of \( a - b \), we can evaluate the approximate values:

- \( \sqrt{8} \approx 2.828 \)
- \( \sqrt{7} \approx 2.646 \)
- \( \sqrt{6} \approx 2.449 \)

Calculating:

\[ a \approx 2.828 - 2.646 \approx 0.182 \]
\[ b \approx 2.646 - 2.449 \approx 0.197 \]

Since \( a \) is approximately \( 0.182 \) and \( b \) is approximately \( 0.197 \), we find that:

\[ a < b \]

### Step 2: Compare \( b \) and \( c \)

Next, we calculate \( b - c \):

\[ b - c = (\sqrt{7} - \sqrt{6}) - (\sqrt{6} - \sqrt{5}) \]
\[ = \sqrt{7} - \sqrt{6} - \sqrt{6} + \sqrt{5} \]
\[ = \sqrt{7} + \sqrt{5} - 2\sqrt{6} \]

Again, we can evaluate the approximate values:

Calculating:

\[ b \approx 0.197 \]
\[ c \approx 2.449 - 2.236 \approx 0.213 \]

Since \( b \) is approximately \( 0.197 \) and \( c \) is approximately \( 0.213 \), we find that:

\[ b < c \]

### Step 3: Conclusion

From the comparisons, we have:

- \( a < b \)
- \( b < c \)

Thus, we can conclude that:

\[ a < b < c \]

### Final Answer

The correct relationship is \( c > b > a \).