The sum of the expression 551 + 552 + 553 + … + 560 is :

Options

A. 3450
B. 5555
C. 555
D. 6060

Answer

To find the sum of the expression \(551 + 552 + 553 + \ldots + 560\), we can recognize that this is an arithmetic progression (AP) where:

1. **First term (A)**: \(551\)
2. **Common difference (D)**: \(552 - 551 = 1\)
3. **Last term**: \(560\)

### Step 1: Determine the number of terms (n)
To find the number of terms in the series, we can use the formula for the \(n\)-th term of an arithmetic progression:

\[ a_n = a + (n-1)d \]

Setting \(a_n = 560\), \(a = 551\), and \(d = 1\):

\[ 560 = 551 + (n-1) \cdot 1 \]

Subtract \(551\) from both sides:

\[ 560 - 551 = n - 1 \]

\[ 9 = n - 1 \]

Adding \(1\) to both sides gives:

\[ n = 10 \]

### Step 2: Use the formula for the sum of an arithmetic series
The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic progression is:

\[ S_n = \frac{n}{2} \cdot (2a + (n-1)d) \]

Substituting the values we found:

- \(n = 10\)
- \(a = 551\)
- \(d = 1\)

We can now substitute these values into the formula:

\[ S_{10} = \frac{10}{2} \cdot (2 \cdot 551 + (10 - 1) \cdot 1) \]

Calculating step-by-step:

1. Calculate \(2a\):

\[ 2 \cdot 551 = 1102 \]

2. Calculate \((n - 1)d\):

\[ (10 - 1) \cdot 1 = 9 \]

3. Add these results together:

\[ 1102 + 9 = 1111 \]

4. Now substitute back into the sum formula:

\[ S_{10} = 5 \cdot 1111 \]

5. Finally, calculate \(5 \cdot 1111\):

\[ 5 \cdot 1111 = 5555 \]

### Final Answer
Thus, the sum of the expression \(551 + 552 + 553 + \ldots + 560\) is:

\[ \boxed{5555} \]