An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the "common difference." A real-life example of an arithmetic sequence could involve saving money.

Real-Life Situation: Saving Money

Scenario:
Imagine you decide to save a certain amount of money every month to buy a new bicycle. You start by saving $50 in the first month, and you plan to increase your savings by $10 each month. How much will you have saved after 12 months?

Step 1: Identify the terms

• First term ($a_1$): This is the amount saved in the first month, $a_1 = 50$.
• Common difference ($d$): The increase in savings each month, $d = 10$.
• Number of terms ($n$): This represents the number of months you're saving, $n = 12$.

Step 2: Write the formula for the $n$th term of an arithmetic sequence

The $n$th term of an arithmetic sequence is given by: $a_n = a_1 + (n-1) \times d$ Where:

• $a_n$ is the amount saved in the nth month.

Step 3: Calculate the savings after 12 months

To find out how much you have saved by the 12th month ($a_{12}$): $a_{12} = 50 + (12-1) \times 10 = 50 + 11 \times 10 = 50 + 110 = 160$ So, in the 12th month, you will save $160.

Step 4: Calculate the total savings after 12 months

The total amount saved after 12 months is the sum of the first 12 terms of the arithmetic sequence. The sum ($S_n$) of the first $n$ terms of an arithmetic sequence is given by: $S_n = \frac{n}{2} \times (a_1 + a_n)$ Substituting the known values: $S_{12} = \frac{12}{2} \times (50 + 160) = 6 \times 210 = 1260$ So, after 12 months, you will have saved a total of $1260.

Summary:

By saving $50 in the first month and increasing your savings by $10 each month, you will have saved $1260 after 12 months.

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Would you like further details or have any questions?

Related Questions:

1. How would the total savings change if the initial savings amount was different?
2. What if the common difference was $5 instead of $10?
3. How would you calculate the amount saved after 6 months instead of 12?
4. How does an arithmetic sequence differ from a geometric sequence?
5. Can you model expenses instead of savings using an arithmetic sequence?
6. How would the sequence change if the monthly increase was negative?
7. What if you skipped saving in one of the months?
8. How can you find the savings after 24 months?

Tip: Remember, the sum formula for arithmetic sequences is useful for quickly finding the total after a specific number of terms, especially when dealing with larger sequences.