To understand how to use the sum formula for arithmetic sequences in word problems, let's first remember what an arithmetic sequence is.

An arithmetic sequence is a list of numbers where the difference between any two numbers next to each other is the same. This difference is called the common difference (we'll call it $d$).

The formula to find the sum of the first $n$ terms ($S_n$) of an arithmetic sequence is:

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

Here's what these symbols mean:

• $S_n$ = the total sum of the first $n$ terms,
• $n$ = the number of terms,
• $a$ = the first term,
• $l$ = the last term.

If you know the common difference, you can also use this formula:

$$S_n = \frac{n}{2} \times (2a + (n-1)d)$$

Now, let’s see how to use this in real life with some examples.

Example Problem 1: Total Cost of Concert Tickets

Imagine you are going to three concerts, and the ticket prices go up by £5 each time. If the first concert ticket costs £20, what will be the total cost for all three concerts?

1. Identify the terms:

   • The first ticket ($a$) is £20.
   • The common difference ($d$) is £5.
   • The number of concerts ($n$) is 3.

2. Find the last term:

   • The price for the third concert ($l$) is $a + 2d = 20 + 2 \times 5 = 30$.

3. Use the sum formula:

   • Let's use the first formula:

$$S_3 = \frac{3}{2} \times (20 + 30) = \frac{3}{2} \times 50 = 75.$$

So, the total cost for all the tickets is £75.

Example Problem 2: Savings Over Time

Now, let’s say you save £10 in the first month. Then, each month you save £5 more. How much will you have saved after 12 months?

1. Identify the terms:

   • The first month's saving ($a$) is £10.
   • The common difference ($d$) is £5.
   • The number of months ($n$) is 12.

2. Find the last term:

   • Your saving in the last month ($l$) is $a + (n - 1)d = 10 + 11 \times 5 = 65$.

3. Use the sum formula:

   • Using the first formula:

$$S_{12} = \frac{12}{2} \times (10 + 65) = 6 \times 75 = 450.$$

So, after 12 months, you will have saved £450.

By breaking down each problem into easy steps, you can use the sum formula for arithmetic sequences in different word problems!