The operation `=rArr` is defined by `x=rArry=x+(x+1)+(x+2)…+y.` For example, `3=rArr7=3+4+5+6+7.` What is the vlaue of `(100=rArr150)-(125=rArr150)`?

Answer

To solve the problem, we need to evaluate the expression \((100 \Rightarrow 150) - (125 \Rightarrow 150)\) using the defined operation \(x \Rightarrow y = x + (x + 1) + (x + 2) + \ldots + y\).

### Step 1: Calculate \(100 \Rightarrow 150\)

Using the definition of the operation, we can express \(100 \Rightarrow 150\) as:

\[ 100 \Rightarrow 150 = 100 + 101 + 102 + \ldots + 150 \]

This is the sum of an arithmetic series where:
- First term \(a = 100\)
- Last term \(l = 150\)
- Number of terms \(n = 150 - 100 + 1 = 51\)

The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic series is:

\[ S_n = \frac{n}{2} \times (a + l) \]

Substituting the values we have:

\[ S_{51} = \frac{51}{2} \times (100 + 150) = \frac{51}{2} \times 250 = 51 \times 125 = 6375 \]

### Step 2: Calculate \(125 \Rightarrow 150\)

Now, we calculate \(125 \Rightarrow 150\):

\[ 125 \Rightarrow 150 = 125 + 126 + 127 + \ldots + 150 \]

This is also an arithmetic series where:
- First term \(a = 125\)
- Last term \(l = 150\)
- Number of terms \(n = 150 - 125 + 1 = 26\)

Using the same formula for the sum:

\[ S_{26} = \frac{26}{2} \times (125 + 150) = 13 \times 275 = 3575 \]

### Step 3: Calculate the difference

Now we need to find the difference:

\[ (100 \Rightarrow 150) - (125 \Rightarrow 150) = 6375 - 3575 = 2800 \]

### Final Answer

Thus, the value of \((100 \Rightarrow 150) - (125 \Rightarrow 150)\) is:

\[ \boxed{2800} \]

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