What is the value of `(1)/(30.31) +(1)/(31.32)+(1)/(32.33) + …+ (1)/(89.90)`?

Options

A. `(1)/(45)`
B. `(2)/(15)`
C. `(2)/(45)`
D. `(2)/(15)`

Answer

To solve the problem, we need to evaluate the sum:

\[ S = \frac{1}{30 \cdot 31} + \frac{1}{31 \cdot 32} + \frac{1}{32 \cdot 33} + \ldots + \frac{1}{89 \cdot 90} \]

### Step 1: Rewrite each term
Each term in the sum can be rewritten using the identity:

\[ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \]

So, we can rewrite the first few terms:

\[ \frac{1}{30 \cdot 31} = \frac{1}{30} - \frac{1}{31} \]
\[ \frac{1}{31 \cdot 32} = \frac{1}{31} - \frac{1}{32} \]
\[ \frac{1}{32 \cdot 33} = \frac{1}{32} - \frac{1}{33} \]

Continuing this pattern, we can express the entire sum \(S\):

\[ S = \left( \frac{1}{30} - \frac{1}{31} \right) + \left( \frac{1}{31} - \frac{1}{32} \right) + \left( \frac{1}{32} - \frac{1}{33} \right) + \ldots + \left( \frac{1}{89} - \frac{1}{90} \right) \]

### Step 2: Simplify the sum
Notice that this is a telescoping series. Most terms will cancel out:

\[ S = \frac{1}{30} - \frac{1}{90} \]

### Step 3: Calculate the remaining terms
Now we need to find a common denominator to combine these two fractions:

The common denominator of 30 and 90 is 90. So we rewrite \(\frac{1}{30}\):

\[ \frac{1}{30} = \frac{3}{90} \]

Now substituting back into the equation for \(S\):

\[ S = \frac{3}{90} - \frac{1}{90} = \frac{2}{90} \]

### Step 4: Simplify the final result
Now simplify \(\frac{2}{90}\):

\[ S = \frac{2}{90} = \frac{1}{45} \]

### Conclusion
Thus, the value of the sum is:

\[ \boxed{\frac{1}{45}} \]