Final answer:
The general term of a binomial expansion can be found using the formula Tn = C(n, k) * an-k * bk. For example, if you are given the sequence 1, 3, 9, 27, ..., the general term can be found by observing the terms and plugging them into the formula.
Step-by-step explanation:
Binomial Theorem:
The general term of a binomial expansion is given by:
Tn = C(n, k) * an-k * bk
Where:
- Tn represents the nth term in the expansion
- C(n, k) is the binomial coefficient given by C(n, k) = n! / (k! * (n-k)!)
- a and b are the terms being added or subtracted
- n is the power to which the binomial is raised
- k represents the term number starting from 0
For example, if you are given the sequence 1, 3, 9, 27, ..., the general term can be found using the formula above.
Tn = C(n, k) * 1n-k * 3k
By observing the terms, we can see that k = 0 for the first term, k = 1 for the second term, k = 2 for the third term, and so on. Therefore, the general term for this sequence is:
Tn = C(n, 0) * 1n-0 * 30
Since C(n, 0) = 1, 1n-0 = 1, and 30 = 1, we have:
Tn = 1 * 1 * 1 = 1
So, the general term for the sequence 1, 3, 9, 27, ... is 1.