Answer :
Final answer:
This response explains how to find the 100th term and the nth term in arithmetic and geometric sequences.
Explanation:
a. Arithmetic sequence:
The common difference between the terms in this sequence is 2. To find the 100th term, we can use the formula:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, n is the term number, and d is the common difference. Plugging in the given values:
a100 = 1 + (100-1)2
a100 = 1 + 198 = 199
Therefore, the 100th term is 199.
To find the nth term, we can use the same formula and plug in the value of n we are given:
an = 1 + (n-1)2
b. Arithmetic sequence:
Here, the common difference between the terms is 40. To find the 100th term, we can use the formula:
a100 = 50 + (100-1)40
a100 = 50 + 99 imes 40 = 50 + 3960 = 4010
Therefore, the 100th term is 4010.
The nth term can be found using the formula:
an = 50 + (n-1)40
c. Geometric sequence:
There is a pattern in this sequence: each term is the previous term squared. The 1st term is 1, the 2nd term is 3, and the 3rd term is 9. To find the nth term, we can use the formula:
an = a1 imes r^(n-1)
where an is the nth term, a1 is the first term, n is the term number, and r is the common ratio. In this case, the common ratio is 3. Plugging in the given values:
an = 1 imes 3^(n-1)
d. Nonlinear sequence:
This sequence does not follow a clear pattern. The terms are not related to each other through a simple equation or rule, so it is not possible to find the nth term or the 100th term with the information provided.
e. Arithmetic sequence:
The common difference between the terms in this sequence is 1. To find the 100th term, we can use the formula:
a100 = 156 + (100-1)1
a100 = 156 + 99 = 255
Therefore, the 100th term is 255.
To find the nth term, we can use the formula:
an = 156 + (n-1)1
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