Question

There are (1)______ ways to find the general rule or nth term of a given sequence. As I went through this topic, I learned that different sequences can be solved by using (2)____(3)_____. I realized that when finding the nth term of a sequence that can't be easily determined by inspection, we need to find its (4)____and/or (5)____differences. ​

Answer 1

There are several ways to find the general rule or nth term of a sequence, including the binomial theorem for series expansions, and examining first and second differences for patterns. Calculators such as the TI-83, 83+, or 84 can aid in complex computations.

Step-by-step explanation:

There are several ways to find the general rule or nth term of a given sequence. Understanding how different sequences operate and the methods to solve them is vital in mathematics. When you can't easily determine the nth term by inspection, it's important to look for patterns in the sequence. One approach is to use the binomial theorem for expansion, which is represented as (a + b)n = an + nan-1b + n(n-1)an-2b2/2! + n(n-1)(n-2)an-3b3/3! + ... and so on.

For sequences that exhibit a pattern of addition or subtraction, you can find the nth term by identifying the first term and the common difference (in arithmetic sequences) or the common ratio (in geometric sequences). However, for more complex sequences, you might need to examine the sequence's first and/or second differences to uncover a pattern. By solving problems step by step, from finding patterns, applying formulas, and using calculators like the TI-83, 83+, or 84 for complex computations.