What is the sum of all numbers of the arithmetic progression with the term 0.23 and common difference 0.1 that stops before the number 10

To find the sum of all the terms in an arithmetic progression (AP) with the given properties, we follow these steps:

### Step 1: Identify the parameters of the arithmetic progression
- **First term (a)**: 0.23
- **Common difference (d)**: 0.1
- **Last term (l)**: The progression stops before 10, meaning the largest term must be less than 10.

### Step 2: Find the number of terms (n)
The general term of an arithmetic progression is given by:

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

We need to find \(n\) such that \(a_n < 10\). Setting \(a_n = 10\) and solving for \(n\):

\[
10 = 0.23 + (n - 1)(0.1)
\]

Simplifying:

\[
10 = 0.23 + 0.1(n - 1)
\]
\[
10 - 0.23 = 0.1(n - 1)
\]
\[
9.77 = 0.1(n - 1)
\]
\[
n - 1 = \frac{9.77}{0.1} = 97.7
\]
\[
n = 98.7
\]

Since \(n\) must be an integer, the number of terms \(n\) is 98 (rounding down because the term at \(n = 99\) would exceed 10).

### Step 3: Use the sum formula for an arithmetic progression
The sum \(S_n\) of the first \(n\) terms of an arithmetic progression is given by:

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

Where:
- \(n = 98\)
- \(a = 0.23\)
- The last term \(l\) is:

\[
l = a + (n - 1)d = 0.23 + (98 - 1)(0.1) = 0.23 + 9.7 = 9.93
\]

Now calculate the sum:

\[
S_{98} = \frac{98}{2} \times (0.23 + 9.93) = 49 \times 10.16 = 497.84
\]

### Final Answer: 
The sum of all the terms in the arithmetic progression is **497.84**.

Would you like more details, or have any questions?

Here are 5 related questions to explore:
1. What is the formula for the nth term of an arithmetic progression?
2. How do you find the sum of a geometric progression?
3. Can you explain how rounding affects solutions in arithmetic progressions?
4. How would the sum change if the common difference were negative?
5. How does the formula for the sum of terms apply to real-life problems?

**Tip:** Always verify whether the last term of an arithmetic sequence exceeds the stopping condition before summing up the terms.

What is the sum of all numbers of the arithmetic progression with the term 0.23 and common difference 0.1 that stops before the number 10?

Find the sum of all terms in an arithmetic progression with a first term of 0.23, common difference of 0.1, and a stopping condition before 10. This problem demonstrates how to use the general term formula and the sum formula for an arithmetic sequence. Suitable for Grades 10-12.

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