Sequence Nth Term: Find Terms & Solve For K

Hey guys! Let's dive into the world of sequences and tackle a problem that involves finding specific terms and solving for an unknown index. Sequences are pretty cool because they're basically ordered lists of numbers, and often, there's a pattern or a rule that generates them. The

nth term of a sequence

is like the master key to this pattern. It's a formula that allows you to calculate any term in the sequence just by plugging in its position number, 'n'. Think of it like a function where 'n' is your input, and the formula gives you the corresponding term's value. Understanding this concept is super important for mastering sequence problems, as it unlocks the ability to predict future terms, find missing ones, and even work backward to figure out the position of a given value. In this article, we’re going to break down a specific problem involving a sequence defined by its nth term, rac{n}{2n+3}. We'll figure out the first few terms and then solve for a specific term's position.

(a) Understanding the Sequence: rac{n}{2n+3}

So, we're given a sequence where the nth term is defined by the formula rac{n}{2n+3}. This means for any position 'n' in the sequence, the value of that term is calculated by taking 'n' and dividing it by '2n + 3'. Pretty straightforward, right? Let's get into the specifics.

(i) Finding the First Three Terms

To find the first three terms of this sequence, we just need to substitute n = 1, n = 2, and n = 3 into the given nth term formula. Remember, the nth term is our roadmap here. Let's go!

• For the 1st term (n=1): We plug in 1 for 'n' in our formula: rac{1}{2(1)+3} = rac{1}{2+3} = rac{1}{5}. So, the first term is rac{1}{5}.
• For the 2nd term (n=2): Now, we plug in 2 for 'n': rac{2}{2(2)+3} = rac{2}{4+3} = rac{2}{7}. The second term is rac{2}{7}.
• For the 3rd term (n=3): Finally, we plug in 3 for 'n': rac{3}{2(3)+3} = rac{3}{6+3} = rac{3}{9}. This fraction can be simplified to rac{1}{3}. So, the third term is rac{1}{3}.

There you have it! The first three terms of the sequence are rac{1}{5}, rac{2}{7}, and rac{1}{3}. It's always good practice to simplify fractions when you can, so rac{3}{9} simplifies nicely to rac{1}{3}. Seeing these terms gives us a better feel for how the sequence is growing. Notice how the denominator increases faster than the numerator, suggesting the terms are getting closer to a certain value (we'll touch on that concept later in more advanced discussions).

(ii) Finding the Value of k

Now, things get a little more interesting. We're told that the kth term of this same sequence is equal to rac{12}{25}. Our mission, should we choose to accept it, is to find the value of 'k'. This means we need to set our nth term formula equal to rac{12}{25} and solve for 'n' (which in this case, we're calling 'k').

The equation we need to solve is:

rac{k}{2k+3} = rac{12}{25}

To solve this, we can use cross-multiplication. This is a standard technique for solving equations involving fractions. We multiply the numerator of the first fraction by the denominator of the second, and set it equal to the product of the denominator of the first fraction and the numerator of the second.

So, we get:

$25 imes k = 12 imes (2k+3)$

Now, let's simplify and solve for 'k':

$25k = 24k + 36$

To isolate 'k', we subtract $24k$ from both sides of the equation:

$25k - 24k = 36$

$k = 36$

And there we have it! The value of k is 36. This means that the 36th term in the sequence is rac{12}{25}. Pretty neat how we can use the nth term formula to find the position of a specific value within a sequence. It shows the power and versatility of these formulas in describing and analyzing numerical patterns.

(b) Finding the nth Term of a New Sequence

This part of the problem is a bit incomplete as it states "Find the nth term of eachDiscussion category : mathematics". It seems like there might be some missing information about which categories or which sequences we need to find the nth term for. However, I can give you a general idea of how you'd approach finding the nth term for different types of sequences, assuming you were given specific examples within those categories.

General Strategies for Finding the nth Term

Finding the nth term of a sequence is all about identifying the pattern. Here are some common types of sequences and how you might find their nth term formulas:

1. Arithmetic Sequences: In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'. The formula for the nth term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and 'd' is the common difference. You'd look for a constant addition or subtraction between terms.

   Example: If a sequence is 3, 7, 11, 15,... The first term ($a_1$) is 3. The common difference (d) is $7-3=4$. So, the nth term is $a_n = 3 + (n-1)4 = 3 + 4n - 4 = 4n - 1$.

2. Geometric Sequences: In a geometric sequence, each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted by 'r'. The formula for the nth term of a geometric sequence is: $a_n = a_1 imes r^{(n-1)}$, where $a_1$ is the first term and 'r' is the common ratio.

   Example: If a sequence is 2, 6, 18, 54,... The first term ($a_1$) is 2. The common ratio (r) is $6/2 = 3$. So, the nth term is $a_n = 2 imes 3^{(n-1)}$.

3. Quadratic Sequences: These sequences have a second difference that is constant. The nth term formula for a quadratic sequence is of the form $a_n = An^2 + Bn + C$. To find A, B, and C, you can use the first few terms and set up a system of equations, or use shortcut formulas based on the differences.

   Example: If a sequence is 1, 4, 9, 16,... This is actually the sequence of perfect squares, so $a_n = n^2$. The second difference is constant (equal to 2). Let's try another: 2, 7, 14, 23,... First differences: 5, 7, 9 Second differences: 2, 2 Since the second difference is 2, A = 2/2 = 1. The formula starts with $1n^2$. Then you find B and C.

4. Sequences with Patterns in Numerators and Denominators: Like the sequence in part (a), sometimes you have separate rules for the numerators and denominators. You'd analyze each part independently to find its nth term formula, and then combine them.

   Example: Sequence: 1/2, 3/4, 5/6, 7/8,... Numerators: 1, 3, 5, 7,... (This is an arithmetic sequence with $a_1=1, d=2$, so the nth term is $1 + (n-1)2 = 2n-1$) Denominators: 2, 4, 6, 8,... (This is an arithmetic sequence with $a_1=2, d=2$, so the nth term is $2 + (n-1)2 = 2n$) So, the nth term of the combined sequence is rac{2n-1}{2n}.

Without specific sequences for part (b), I can't give you a definitive answer. However, the key takeaway is to always look for a pattern. Calculate the differences between terms, look for constant ratios, or see if the numerators and denominators follow their own predictable rules. Once you spot the pattern, you can formulate the nth term that describes it. Keep practicing, guys, and you'll become sequence wizards in no time! Understanding the nth term is fundamental, and with a bit of practice, you'll be able to crack any sequence puzzle thrown your way.