Find the sum of all two-digit natural numbers that are multiples of 5




To find the sum of all two-digit natural numbers that are multiples of 5:

### Step 1: Identify the two-digit multiples of 5
The smallest two-digit natural number that is a multiple of 5 is \(10\).  
The largest two-digit natural number that is a multiple of 5 is \(95\).

Thus, the sequence of two-digit multiples of 5 is:
\[
10, 15, 20, 25, \dots, 95
\]

This is an **arithmetic sequence** where:
- The first term (\(a\)) is \(10\),
- The common difference (\(d\)) is \(5\),
- The last term (\(l\)) is \(95\).

### Step 2: Determine the number of terms in the sequence
The formula for the \(n\)-th term of an arithmetic sequence is:
\[
a_n = a + (n-1)d
\]

Set \(a_n = 95\), \(a = 10\), and \(d = 5\):
\[
95 = 10 + (n-1) \cdot 5
\]
\[
95 - 10 = (n-1) \cdot 5
\]
\[
85 = (n-1) \cdot 5
\]
\[
n-1 = 17 \implies n = 18
\]

Thus, there are \(18\) terms in the sequence.

### Step 3: Calculate the sum of the sequence
The sum of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} \cdot (a + l)
\]

Substitute \(n = 18\), \(a = 10\), and \(l = 95\):
\[
S_{18} = \frac{18}{2} \cdot (10 + 95)
\]
\[
S_{18} = 9 \cdot 105
\]
\[
S_{18} = 945
\]

### Final Answer:
The sum of all two-digit natural numbers that are multiples of 5 is:
\[
\boxed{945}
\]

---

Would you like further details or have any other questions?  
Here are 5 related questions for further exploration:

1. How do you calculate the sum of multiples of any number within a range?
2. What if the problem involved three-digit multiples of 5 instead?
3. Can you explain how to find the number of terms in any arithmetic sequence?
4. How would this change if the numbers had to be divisible by both 5 and another number (e.g., 3)?
5. How can this method be applied to solve problems involving geometric sequences?

**Tip:** Always check for the sequence type (arithmetic, geometric, etc.) to use the right summation formula efficiently.

Find the sum of all two-digit natural numbers that are multiples of 5.

Find the sum of all two-digit natural numbers that are multiples of 5 using arithmetic sequences. This detailed solution includes identifying terms, calculating the number of terms, and summing the sequence using key formulas. Perfect for students in Grades 6-8.

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