Simplify : `1'2/(15) + 2'2/(15) + 3'2/(15) + "……" + 100'2/(15)`

Explanation

To simplify the expression \(1\frac{2}{15} + 2\frac{2}{15} + 3\frac{2}{15} + \ldots + 100\frac{2}{15}\), we can follow these steps:

### Step 1: Rewrite the Mixed Numbers
First, we rewrite each mixed number as an improper fraction.

\[ 1\frac{2}{15} = \frac{15 \times 1 + 2}{15} = \frac{17}{15} \]
\[ 2\frac{2}{15} = \frac{15 \times 2 + 2}{15} = \frac{32}{15} \]
\[ 3\frac{2}{15} = \frac{15 \times 3 + 2}{15} = \frac{47}{15} \]
\[ \vdots \]
\[ 100\frac{2}{15} = \frac{15 \times 100 + 2}{15} = \frac{1502}{15} \]

### Step 2: Factor Out the Common Denominator
Notice that all terms have a common denominator of 15. We can factor this out:

\[ \frac{1}{15} \left( 17 + 32 + 47 + \ldots + 1502 \right) \]

### Step 3: Identify the Pattern
The numerators form an arithmetic series where the first term \(a = 17\) and the last term \(l = 1502\). The common difference \(d = 15\).

### Step 4: Find the Number of Terms
To find the number of terms \(n\) in this arithmetic series, we can use the formula for the \(n\)-th term of an arithmetic series:

\[ l = a + (n-1)d \]
Substituting the known values:

\[ 1502 = 17 + (n-1) \times 15 \]
\[ 1502 - 17 = (n-1) \times 15 \]
\[ 1485 = (n-1) \times 15 \]
\[ n-1 = \frac{1485}{15} = 99 \]
\[ n = 100 \]

### Step 5: Calculate the Sum of the Series
The sum \(S_n\) of the first \(n\) terms of an arithmetic series can be calculated using the formula:

\[ S_n = \frac{n}{2} \times (a + l) \]
Substituting the values we found:

\[ S_{100} = \frac{100}{2} \times (17 + 1502) = 50 \times 1519 = 75950 \]

### Step 6: Final Calculation
Now we substitute back into our expression:

\[ \frac{1}{15} \times 75950 = \frac{75950}{15} \]

### Step 7: Simplify the Fraction
Now we simplify \(\frac{75950}{15}\):

\[ 75950 \div 15 = 5063.33 \text{ or } 5063 \frac{1}{3} \]

### Final Answer
Thus, the simplified form of the expression is:

\[ 5063 \frac{1}{3} \]

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