Figuring Out the Sequence: Finding the Term That's 84 More Than the 13th
Ever looked at a list of numbers and wondered about the pattern? We're diving into a common type of math sequence called an arithmetic progression, or AP for short. In this article, we'll tackle a specific question: Which term of the AP 3, 10, 17 will be 84 more than its 13th term? We'll break down how to solve this step-by-step, so even if math isn't your favorite subject, you can follow along and understand the logic.
Understanding Arithmetic Progressions
An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the "common difference."
Let's look at our AP: 3, 10, 17.
- The first term ($a_1$) is 3.
- To find the common difference ($d$), we subtract any term from its succeeding term:
- $10 - 3 = 7$
- $17 - 10 = 7$
- So, the common difference ($d$) for this AP is 7.
The Formula for an Arithmetic Progression
To find any term in an arithmetic progression, we use a handy formula:
$a_n = a_1 + (n-1)d$
Where:
- $a_n$ is the $n$-th term we want to find.
- $a_1$ is the first term.
- $n$ is the position of the term in the sequence (like 1st, 2nd, 3rd, etc.).
- $d$ is the common difference.
Step 1: Finding the 13th Term
Before we can find a term that's 84 more than the 13th term, we first need to figure out what the 13th term actually is. We'll use our formula with the information we have:
- $a_1 = 3$
- $d = 7$
- $n = 13$ (because we're looking for the 13th term)
Plugging these values into the formula:
$a_{13} = 3 + (13-1) * 7$
$a_{13} = 3 + (12) * 7$
$a_{13} = 3 + 84$
$a_{13} = 87$
So, the 13th term of the AP 3, 10, 17 is 87.
Step 2: Determining the Target Term
The question asks for the term that will be 84 *more than* its 13th term. We just found that the 13th term is 87.
Therefore, the term we're looking for is:
Target Term = 13th Term + 84
Target Term = 87 + 84
Target Term = 171
Our goal is to find which term in the sequence will have the value of 171.
Step 3: Finding the Position ($n$) of the Target Term
Now we know the value of the term we're looking for ($a_n = 171$). We can use our arithmetic progression formula again, but this time we'll be solving for '$n$', the position of the term.
We have:
- $a_n = 171$ (the value of the term we want)
- $a_1 = 3$ (the first term)
- $d = 7$ (the common difference)
- $n = ?$ (this is what we need to find)
Let's plug these into the formula and solve for $n$:
$171 = 3 + (n-1) * 7$
Now, we isolate $n$:
$171 - 3 = (n-1) * 7$
$168 = (n-1) * 7$
To get $(n-1)$ by itself, we divide both sides by 7:
$168 / 7 = n-1$
$24 = n-1$
Finally, to find $n$, we add 1 to both sides:
$24 + 1 = n$
$n = 25$
Conclusion
By following these steps, we've determined that the 25th term of the arithmetic progression 3, 10, 17 will be 84 more than its 13th term.
Let's quickly check our work:
- We found the 13th term is 87.
- We need a term that is 87 + 84 = 171.
- We found the 25th term ($a_{25}$). Let's calculate it:
- $a_{25} = 3 + (25-1) * 7$
- $a_{25} = 3 + (24) * 7$
- $a_{25} = 3 + 168$
- $a_{25} = 171$
Our calculations are correct!
Frequently Asked Questions (FAQ)
How do I find the common difference in an arithmetic progression?
To find the common difference, simply subtract any term from the term that comes immediately after it. For example, in the sequence 3, 10, 17, you can do 10 - 3 = 7, or 17 - 10 = 7. The result will be the same for all consecutive terms.
Why is the formula $a_n = a_1 + (n-1)d$ used?
This formula works because each subsequent term is found by adding the common difference ($d$) to the previous term. For the second term ($n=2$), you add $d$ once to $a_1$. For the third term ($n=3$), you add $d$ twice to $a_1$. In general, for the $n$-th term, you add $d$ a total of $(n-1)$ times to the first term ($a_1$).
What if I'm asked for a term that's less than the 13th term?
The process is the same. If you needed a term that was, say, 84 *less* than the 13th term, you would calculate the 13th term (which is 87) and then subtract 84 from it (87 - 84 = 3). Then you would use the formula to find which term in the sequence equals 3, which we already know is the first term.
