Question

Homework

Answer 1

Note: As you have missed to mention the first four terms of the Arithmetic sequence. So, I am randomly assuming that first four terms of the arithmetic sequence be 1, 3, 5, 7... This would anyhow make you understand the concept. So, I am solving your query based on assuming the first four terms of an Arithmetic sequence as 1, 3, 5, 7...

Part A)

What is the next term of this sequence?

Answer:

${\displaystyle \ a_{5}=9$ is the next term i.e. 5th term of the arithmetic sequence 1, 3, 5, 7...

Step-by-step explanation:

Considering the Arithmetic sequence with fist four terms

1, 3, 5, 7...

As we know that a sequence is termed as arithmetic sequence of numbers if the difference of any two consecutive terms of the sequence remains constant.

For instance, 1, 3, 5, 7... will be an arithmetic sequence having the common difference 2. Common difference is denoted by 'd'.

So,

Given the sequence

1, 3, 5, 7...

$d=3-1=2,d=5-3=2$

As $a_{1} = 1$ and $d = 2$

The next term i.e. 5th term can be found by using the $nth$ term of the sequence.

So, consider the nth term of the sequence ${\displaystyle a_{n}$

${\displaystyle \ a_{n}=a_{1}+(n-1)d}$

Putting $n=5$ in, $a_{1} = 1$ and $d = 2$  in ${\displaystyle \ a_{n}=a_{1}+(n-1)d}$ to find the 5th term.

${\displaystyle \ a_{n}=a_{1}+(n-1)d}$

${\displaystyle \ a_{5}=1+(5-1)2}$

${\displaystyle \ a_{5}=1+(4)2}$

${\displaystyle \ a_{5}=9$

So, ${\displaystyle \ a_{5}=9$ is the next term i.e. 5th term of the arithmetic sequence 1, 3, 5, 7...

Part B)

Writing down an expression,  in terms of n for the nth term of the sequence

consider the nth term of the sequence ${\displaystyle a_{n}$

${\displaystyle \ a_{n}=a_{1}+(n-1)d}$

Here, $a_{1}$ is the first term, $d$ is the common difference.

For example,

Given the sequence

1, 3, 5, 7...

$d=3-1=2,d=5-3=2$

As $a_{1} = 1$ and $d = 2$

The next term i.e. 5th term can be found by using the $nth$ term of the sequence.

So, consider the nth term of the sequence ${\displaystyle a_{n}$

${\displaystyle \ a_{n}=a_{1}+(n-1)d}$

Putting $n=5$ in, $a_{1} = 1$ and $d = 2$  in ${\displaystyle \ a_{n}=a_{1}+(n-1)d}$ to find the 5th term.

${\displaystyle \ a_{n}=a_{1}+(n-1)d}$

${\displaystyle \ a_{5}=1+(5-1)2}$

${\displaystyle \ a_{5}=1+(4)2}$

${\displaystyle \ a_{5}=9$

Keywords: arithmetic sequence, nth term, common difference

Learn more abut arithmetic sequence, nth term and common difference from brainly.com/question/12227567

#learnwithBrainly