Question

For the sequence a_n=3/n+1Find 1st term:Find Second Term:Find third term:Find fourth term:find 100th term:

Answer 1

Given:

The sequence is

`a_n=(3)/(n+1)`

Required:

Find the first term, the second term, the third term, the fourth term and the 100th term.

Step-by-step explanation:

The given sequence is:

`a_(n)=(3)/(n+1)`

Substitute n = 1

`\begin{gathered} a_1=(3)/(1+1) \\ a_1=(3)/(2) \end{gathered}`

Substitute n = 2

`\begin{gathered} a_2=(3)/(2+1) \\ a_2=(3)/(3) \\ a_2=1 \end{gathered}`

Substitute n = 3

`\begin{gathered} a_3=(3)/(3+1) \\ a_3=(3)/(4) \end{gathered}`

Substitute n = 4

`\begin{gathered} a_4=(3)/(4+1) \\ a_4=(3)/(5) \end{gathered}`

The term of the sequence are:

`(3)/(2),1,(3)/(4),(3)/(5)`

The given series is in HP

We will write it in AP as:

`(2)/(3),1,(4)/(3),(5)/(3)`

So the common difference of the given sequence is:

`\begin{gathered} 1-(2)/(3)=(1)/(3) \\ (4)/(3)-1=(1)/(3) \\ (5)/(3)-(4)/(3)=(1)/(3) \end{gathered}`

The nth term of the AP series is given by the formula:

`a_n=a+(n-1)d`

where a = first term

n = number of terms

d = common difference

`\begin{gathered} a_(100)=(2)/(3)+(100-1)*(1)/(3) \\ a_(100)=(2)/(3)+99*(1)/(3) \\ a_(100)=(2+99)/(3) \\ a_(100)=(101)/(3) \end{gathered}`

This is the 100th term for the AP.

The 100th term of the given HP sequence is:

`(3)/(101)`

Final Answer:

`\begin{gathered} First\text{ term = }(3)/(2) \\ Second\text{ term = 1} \\ Third\text{ term = }(3)/(4) \\ Fourth\text{ term = }(3)/(5) \\ 100th\text{ term = }(3)/(101) \end{gathered}`