Question

How can each type of sequence be helpful in everyday situations? Provide an example and the corresponding equation for each type.

Answer 1

The two types of the sequence include

1) Arithmetic sequence

2) Geometric sequence

An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term. For example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6. An arithmetic sequence can be known as an arithmetic progression.

The Nth term of an arithmetic sequence is given below as

`\begin{gathered} T_n=a+(n-1)d \\ \text{Where,} \\ a=\text{first term} \\ n=Num\text{mber of terms} \\ d=\text{common difference}=T_2-T_1=T_3-T_2 \end{gathered}`

A geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

The Nth term of a geometric sequence is given below as

`\begin{gathered} U_n=ar^(n-1) \\ \text{Where,} \\ a=\text{first term} \\ r=\text{common ratio=}(t_2)/(t_1)=(t_3)/(t_2) \end{gathered}`

Real life illustrations of the types of sequence

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 boy 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 6 rows in the tower. Find a for n = 6

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.

`\begin{gathered} t_n=a+(n-1)d \\ T_6=15+(6-1)(-2) \\ T_6=15+5(-2) \\ T_6=15-10 \\ T_6=5 \end{gathered}`

Hence,

On the sixth row, he will have 5 blocks left

Example: geometric sequence

A country's 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?

Here,the common ratio r=1.5

`U_n=a(1.5)^(n-1)`