Sure! Let’s break down how to make tricky word problems about sequences and series easier to understand. When you see these kinds of problems, the best way to solve them is to take it one step at a time. Here are some simple steps you can follow:

Step 1: Understand the Problem

First, read the problem carefully.

Try to figure out what it’s asking you to do.

Underline or highlight important details.

For example, look for clues that show a sequence or series, like “every third term” or “sum of the first n terms.”

Example: If a problem says, "The first term of a sequence is 2, and each next term goes up by 5," note that this is an arithmetic sequence.

Step 2: Identify the Type of Sequence or Series

Next, figure out what kind of sequence or series you have.

• Arithmetic Sequence: This means you get each term by adding a set number (called the common difference).

• Geometric Sequence: Here, you get each term by multiplying by a set number (called the common ratio).

Example: If we continue with our earlier problem, the sequence looks like $2, 7, 12, 17, \ldots$ The common difference is $5$.

Step 3: Write Down Known Values

Make a list of what you know. This includes the first term and, for arithmetic sequences, the common difference or, for geometric sequences, the common ratio.

• For our arithmetic sequence:
  • First term ($a_1$): 2
  • Common difference ($d$): 5

Step 4: Use Formulas

Use the right formulas for solving the sequence or series.

Make sure you know what you need to find out.

Arithmetic Sequence Formula:

To find the nth term: $a_n = a_1 + (n-1)d$

Arithmetic Series Formula:

To find the sum of the first n terms: $S_n = \frac{n}{2}(a_1 + a_n)$

Geometric Sequence Formula:

To find the nth term: $a_n = a_1 \cdot r^{(n-1)}$

Geometric Series Formula:

To find the sum of the first n terms: $S_n = a_1 \frac{1-r^n}{1-r}$ (when $r \neq 1$)

Step 5: Solve Step-by-Step

Now, solve the problem step-by-step using the formulas.

Example: Let's find the sum of the first 5 terms of the arithmetic sequence $2, 7, 12, 17, \ldots$.

1. Identify $a_1 = 2$ and $d = 5$.
2. Find the 5th term: $a_5 = 2 + (5-1) \cdot 5 = 2 + 20 = 22$
3. Calculate the sum: $S_5 = \frac{5}{2}(2 + 22) = \frac{5}{2}(24) = 60$

Step 6: Double Check Your Work

Finally, when you’re done, make sure to check your work.

Go through each step again to see if everything makes sense, and check your math for any mistakes.

By following these steps, you can make sense of tricky word problems with sequences and series.

Take your time, and you’ll see that these problems can be much easier to handle!