The sum of integers from 113 to 113113 which are divisible by 7 is :

Options

A. a. `92358576`
B. b. `913952088`
C. c. `94501895`
D. d. `912952066`

Answer

To find the sum of integers from 113 to 113113 that are divisible by 7, we can follow these steps:

### Step 1: Identify the first and last terms divisible by 7
- The first integer greater than or equal to 113 that is divisible by 7 can be found by calculating \( 113 \div 7 \) and rounding up to the nearest whole number, then multiplying by 7.

\[ \text{First term} = 7 \times \lceil \frac{113}{7} \rceil = 7 \times 17 = 119 \]

- The last integer less than or equal to 113113 that is divisible by 7 can be found by calculating \( 113113 \div 7 \) and rounding down to the nearest whole number, then multiplying by 7.

\[ \text{Last term} = 7 \times \lfloor \frac{113113}{7} \rfloor = 7 \times 16159 = 113113 \]

### Step 2: Determine the number of terms in the sequence
- The numbers divisible by 7 from 119 to 113113 form an arithmetic progression (AP) where:
- First term \( a = 119 \)
- Last term \( l = 113113 \)
- Common difference \( d = 7 \)

- To find the number of terms \( n \) in the AP, we use the formula for the \( n \)-th term of an AP:

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

Rearranging gives:

\[ n = \frac{l - a}{d} + 1 \]

Substituting the values:

\[ n = \frac{113113 - 119}{7} + 1 = \frac{113113 - 119}{7} + 1 = \frac{113113 - 119}{7} + 1 = \frac{113113 - 119}{7} + 1 = \frac{113113 - 119}{7} + 1 = \frac{113113 - 119}{7} + 1 = \frac{113113 - 119}{7} + 1 = \frac{113113 - 119}{7} + 1 = 16143 \]

### Step 3: Calculate the sum of the arithmetic series
- The sum \( S_n \) of the first \( n \) terms of an arithmetic series can be calculated using the formula:

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

- Substituting the values we found:

\[ S_{16143} = \frac{16143}{2} \cdot (119 + 113113) \]

\[ S_{16143} = \frac{16143}{2} \cdot 113232 \]

\[ S_{16143} = 16143 \cdot 56616 = 913952088 \]

### Final Answer
The sum of integers from 113 to 113113 that are divisible by 7 is **913952088**.

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