<u><em>Note:</em></u><em> 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...</em>
Part A)
<em><u>What is the next term of this sequence?</u></em>
Answer:
is the next term i.e. 5th term of the arithmetic sequence <em>1, 3, 5, 7...</em>
Step-by-step explanation:
Considering the Arithmetic sequence with fist four terms
<em> 1, 3, 5, 7...</em>
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, <em> 1, 3, 5, 7... </em>will be an arithmetic sequence having the common difference 2. Common difference is denoted by 'd'.
So,
Given the sequence
<em>1, 3, 5, 7...</em>
![d=3-1=2,d=5-3=2](https://tex.z-dn.net/?f=d%3D3-1%3D2%2Cd%3D5-3%3D2)
As
and ![d = 2](https://tex.z-dn.net/?f=d%20%3D%202)
The next term i.e. 5th term can be found by using the
term of the sequence.
So, consider the nth term of the sequence
![{\displaystyle \ a_{n}=a_{1}+(n-1)d}](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7Bn%7D%3Da_%7B1%7D%2B%28n-1%29d%7D)
Putting
in,
and
in
to find the 5th term.
![{\displaystyle \ a_{n}=a_{1}+(n-1)d}](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7Bn%7D%3Da_%7B1%7D%2B%28n-1%29d%7D)
![{\displaystyle \ a_{5}=1+(5-1)2}](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7B5%7D%3D1%2B%285-1%292%7D)
![{\displaystyle \ a_{5}=1+(4)2}](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7B5%7D%3D1%2B%284%292%7D)
![{\displaystyle \ a_{5}=9](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7B5%7D%3D9)
So,
is the next term i.e. 5th term of the arithmetic sequence <em>1, 3, 5, 7...</em>
Part B)
<u><em>Writing down an expression, in terms of n for the nth term of the sequence</em></u>
consider the nth term of the sequence
![{\displaystyle \ a_{n}=a_{1}+(n-1)d}](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7Bn%7D%3Da_%7B1%7D%2B%28n-1%29d%7D)
Here,
is the first term,
is the common difference.
For example,
Given the sequence
<em>1, 3, 5, 7...</em>
![d=3-1=2,d=5-3=2](https://tex.z-dn.net/?f=d%3D3-1%3D2%2Cd%3D5-3%3D2)
As
and ![d = 2](https://tex.z-dn.net/?f=d%20%3D%202)
The next term i.e. 5th term can be found by using the
term of the sequence.
So, consider the nth term of the sequence
![{\displaystyle \ a_{n}=a_{1}+(n-1)d}](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7Bn%7D%3Da_%7B1%7D%2B%28n-1%29d%7D)
Putting
in,
and
in
to find the 5th term.
![{\displaystyle \ a_{n}=a_{1}+(n-1)d}](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7Bn%7D%3Da_%7B1%7D%2B%28n-1%29d%7D)
![{\displaystyle \ a_{5}=1+(5-1)2}](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7B5%7D%3D1%2B%285-1%292%7D)
![{\displaystyle \ a_{5}=1+(4)2}](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7B5%7D%3D1%2B%284%292%7D)
![{\displaystyle \ a_{5}=9](https://tex.z-dn.net/?f=%7B%5Cdisplaystyle%20%5C%20a_%7B5%7D%3D9)
Keywords: arithmetic sequence, nth term, common difference
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