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: $\boxed{945}$

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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.