To find the sum of all two-digit natural numbers that leave a remainder of 5 when divided by 7, we can follow these steps:
### Step 1: Identify the first two-digit number
The first two-digit number that leaves a remainder of 5 when divided by 7 can be found by checking the smallest two-digit number, which is 10.
When we divide 10 by 7, the remainder is 3.
Continuing this process:
- 11 gives a remainder of 4
- 12 gives a remainder of 5
Thus, the first two-digit number is **12**.
### Step 2: Identify the last two-digit number
Next, we need to find the largest two-digit number that leaves a remainder of 5 when divided by 7. The largest two-digit number is 99.
When we divide 99 by 7, we find:
- 99 ÷ 7 = 14 remainder 0
- 98 gives a remainder of 6
- 97 gives a remainder of 5
Thus, the last two-digit number is **97**.
### Step 3: Form the sequence
The two-digit numbers that leave a remainder of 5 when divided by 7 form an arithmetic sequence:
- First term (a) = 12
- Last term (l) = 97
- Common difference (d) = 7 (since we are adding 7 to each term)
The sequence is: 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89, 96.
### Step 4: Find the number of terms (n)
To find the number of terms in the sequence, we can use the formula for the nth term of an arithmetic sequence:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 97 = 12 + (n - 1) \cdot 7 \]
Rearranging gives:
\[ 97 - 12 = (n - 1) \cdot 7 \]
\[ 85 = (n - 1) \cdot 7 \]
\[ n - 1 = \frac{85}{7} \]
\[ n - 1 = 12.14 \] (not an integer, so we need to check our last term)
We can also check:
The last term we found is 96, so:
\[ 96 = 12 + (n - 1) \cdot 7 \]
\[ 96 - 12 = (n - 1) \cdot 7 \]
\[ 84 = (n - 1) \cdot 7 \]
\[ n - 1 = \frac{84}{7} \]
\[ n - 1 = 12 \]
\[ n = 13 \]
So, there are **13 terms**.
### Step 5: Calculate the sum (S)
The sum of the first n terms of an arithmetic sequence can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the known values:
\[ S_{13} = \frac{13}{2} \cdot (12 + 96) \]
\[ S_{13} = \frac{13}{2} \cdot 108 \]
\[ S_{13} = 13 \cdot 54 \]
\[ S_{13} = 702 \]
Thus, the sum of all two-digit natural numbers which leave a remainder of 5 when divided by 7 is **702**.
