91Ó°ÊÓ

In mathematics, a sequence is an ordered list of numbers following a particular pattern. Each number in the sequence is called a term. To describe a sequence, we often use something called the "general term." The general term is a formula that allows us to find the terms in the sequence based on their positions. For example, consider the sequence with the general term \( a_n = \frac{1}{7-n} \). Here, the position of each term is represented by \( n \). This equation tells us exactly how to calculate any term we are interested in by just plugging in the position we need.Finding the general term of a sequence is crucial if you want to determine specific entries without having to manually plot each one by hand.

Once you have a general term formula like \( a_n = \frac{1}{7-n} \), the next step to figure out specific terms in your sequence is to perform term substitution. This involves substituting values into \( n \) to find the actual terms in the sequence. Here’s how this process works:

• Start with \( n=1 \). Substitute it into the general formula: \( a_1 = \frac{1}{7-1} = \frac{1}{6} \). Thus, the first term is \( \frac{1}{6} \).
• Next, try \( n=2 \). Substitute this value: \( a_2 = \frac{1}{7-2} = \frac{1}{5} \). The second term is \( \frac{1}{5} \).

Repeat this process with subsequent values of \( n \) to get more terms. This makes term substitution straightforward provided the general term is known.

Once you've got the hang of substituting terms using your general formula, sequence calculation is just connecting the dots. It involves computing terms systematically.To calculate a sequence, simply apply substitution for as many terms as needed. For instance, continuing our sequence:

• When \( n=3 \), we get \( a_3 = \frac{1}{7-3} = \frac{1}{4} \).
• For \( n=4 \), calculate \( a_4 = \frac{1}{7-4} = \frac{1}{3} \).
• At \( n=5 \), you'll find \( a_5 = \frac{1}{7-5} = \frac{1}{2} \).

So, the first five terms of this sequence are \( \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{1}{2} \). Each step builds on the previous, illustrating the power of a general term in sequence calculation.