Answers
Part 1
Arithmetic sequence is a sequence by which the next term is found by adding a constant number. It can be a positive number or a negative number. This number is called the common difference. On the other hand, a geometric sequence is one whose next term is found by multiplying the previous term with a constant (common ratio).
Part 2
Sequences are useful in our daily lives as well as in higher mathematics. For example, the interest portion of monthly payments made to pay off an automobile or home loan, and the list of maximum daily temperatures in one area for a month are sequences.
Example: arithmetic sequence
A child building a tower with blocks uses 15 for the bottom row. Each row has 2 fewer blocks than the previous row. Suppose that there are 8 rows in the tower. Find an for n = 8.
The number of blocks in each row forms an arithmetic sequence with a₁ = 15 and d= −2. The formula to be used is an = a₁ + (n − 1)d.
Example: geometric sequence
An insect population is growing in such a way that each new generation is 1.5 times as large as the previous generation. Suppose there are 100 insects in the first generation. How many will there be in the fifth generation?
The population can be written as a geometric sequence with a₁ as the first generation population, a₂ as the second-generation population, and so on. Then the fifth generation population will be a₅. The formula to be used is an = a₁×r⁽ⁿ⁻¹⁾
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