# How to Mastering Sequence Word Problems: A Comprehensive Guide

Understanding word problems involving sequences requires a step-by-step approach. Here's a guide to help you navigate these problems effectively:

## Step-by-step Guide to Mastering Sequence Word Problems

### Step 1: Identify the Type of Sequence

Word problems involving sequences typically deal with arithmetic or geometric sequences.

• Arithmetic sequences have a constant difference between consecutive terms. For example, in the sequence $$2, 5, 8, 11$$, …, each term is $$3$$ more than the previous one.
• Geometric sequences have a constant ratio between consecutive terms. For example, in the sequence $$3, 6, 12, 24$$, …, each term is twice the previous one.

### Step 2: Understand the Problem

Read the problem carefully and try to understand what is being asked. Look for clues that indicate whether the sequence is arithmetic or geometric.

### Step 3: Identify Known Elements

Determine which elements of the sequence are given and which you need to find. This may include the first term, the common difference or ratio, a particular term in the sequence, or the sum of a certain number of terms.

### Step 4: Formulate Equations

Based on the identified elements, formulate equations.

• For an arithmetic sequence, the $$n$$th term is given by $$a_n​=a_1​+(n−1)d$$, where $$a_1​$$ is the first term and $$d$$ is the common difference.
• For a geometric sequence, the $$n$$th term is $$a_n​=a_1​×r^{(n−1)}$$, where $$a_1​$$ is the first term and $$r$$ is the common ratio.

### Step 5: Solve the Equations

Use the equations to solve for the unknowns. This might involve solving for $$n$$, $$a_n​$$​, $$d$$, or $$r$$, depending on the problem.

### Step 6: Consider Special Conditions

Sometimes, word problems may include special conditions like the sum of a certain number of terms. For these cases:

• The sum of the first $$n$$ terms of an arithmetic sequence is given by $$S_n​=\frac{n​}{2}(a_1​+a_n​)$$ or $$S_n​=\frac{n​}{2}​(2a_1​+(n−1)d$$).
• The sum of the first $$n$$ terms of a geometric sequence is $$S_n​=a_1​\frac{1−r^n}{1−r}$$​ for $$r≠1$$.

Always recheck your solution to ensure it makes sense in the context of the problem. Verify that your answer adheres to the sequence’s rules and the problem’s specifics.

### Step 8: Practice Regularly

Regular practice with different types of sequence word problems enhances understanding and problem-solving skills.

By following these steps, you can systematically approach and solve word problems involving sequences, making them less daunting and more manageable.

### Examples:

Example 1:

A staircase has a total of $$15$$ steps. The first step is $$2$$ inches high, and each successive step is $$1$$ inch higher than the previous one. How high is the $$15$$th step?

Solution:
This problem describes an arithmetic sequence where each step’s height increases by a constant amount. The first term $$a_1$$​ is $$2$$ inches (height of the first step), and the common difference $$d$$ is $$1$$ inch (the increase in height from one step to the next).

We need to find the height of the $$15$$th step, which is the $$15$$th term of the sequence ($$a_{15}$$​).

Using the formula for the $$n$$th term of an arithmetic sequence: $$a_n​=a_1​+(n−1)d$$

Substitute the given values: $$a_{15​}=2+(15−1)×1=2+14×1=2+14=16$$

So, the height of the $$15$$th step is $$16$$ inches.

Example 2:

A certain type of bacteria doubles in number every hour. If there is initially one bacterium, how many bacteria will there be after $$6$$ hours?

Solution:
This problem describes a geometric sequence where the quantity of bacteria is multiplied by a constant factor at each stage. The first term $$a_1$$​ is $$1$$ (initial bacterium), and the common ratio $$r$$ is $$2$$ (since the bacteria double each hour).

We need to find the number of bacteria after $$6$$ hours, which is the $$6$$th term of the sequence ($$a_6$$​).

Using the formula for the $$n$$th term of a geometric sequence: $$a_n​=a_1​×r^{(n−1)}$$

Substitute the given values: $$a_6​=1×2^{(6−1)}=1×2^{5}=1×32=32$$

So, there will be $$32$$ bacteria after 6 hours.

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