Question
What is the general formula for an arithmetic sequence, and can you give an example of its application in daily life?
Answers
GPT-4o mini
The general formula for an arithmetic sequence can be expressed as:
\[ a_n = a_1 + (n - 1) \cdot d \]
Where:
- \( a_n \) is the nth term of the sequence,
- \( a_1 \) is the first term,
- \( d \) is the common difference between consecutive terms, and
- \( n \) is the term number.
### Example in Daily Life
Consider a situation where you save a fixed amount of money every week. Suppose you start by saving $10 in the first week and you decide to increase that amount by $5 every subsequent week.
In this case:
- \( a_1 = 10 \) (savings in the first week),
- \( d = 5 \) (the increase in savings every week).
Using the arithmetic sequence formula, the amount saved in week \( n \) would be:
\[ a_n = 10 + (n - 1) \cdot 5 \]
So if you want to find out how much you will save in the 4th week:
\[ a_4 = 10 + (4 - 1) \cdot 5 \]
\[ a_4 = 10 + 3 \cdot 5 \]
\[ a_4 = 10 + 15 = 25 \]
Therefore, you will save $25 in the 4th week.
This simple model shows how arithmetic sequences can help you plan savings over time and project how much you will have saved by a certain week.
\[ a_n = a_1 + (n - 1) \cdot d \]
Where:
- \( a_n \) is the nth term of the sequence,
- \( a_1 \) is the first term,
- \( d \) is the common difference between consecutive terms, and
- \( n \) is the term number.
### Example in Daily Life
Consider a situation where you save a fixed amount of money every week. Suppose you start by saving $10 in the first week and you decide to increase that amount by $5 every subsequent week.
In this case:
- \( a_1 = 10 \) (savings in the first week),
- \( d = 5 \) (the increase in savings every week).
Using the arithmetic sequence formula, the amount saved in week \( n \) would be:
\[ a_n = 10 + (n - 1) \cdot 5 \]
So if you want to find out how much you will save in the 4th week:
\[ a_4 = 10 + (4 - 1) \cdot 5 \]
\[ a_4 = 10 + 3 \cdot 5 \]
\[ a_4 = 10 + 15 = 25 \]
Therefore, you will save $25 in the 4th week.
This simple model shows how arithmetic sequences can help you plan savings over time and project how much you will have saved by a certain week.