Question

Find the 5th term in a sequence whose general term is an=3n+7A)8B)22C)17D)25

Answer 1

There are two basic types of sequences that have generalized forms.

Arithmatic sequences:- This type deals with a sequence of terms that are separated by arithmetic operations ( addition / subtraction ). Either the next term in the sequence is greater than precceding term ( addition of common difference ) or lesser than preceeding term ( subtraction ).

E.g:-

1 , 2 , 3 , 4 , 5 , 6 , 7 , ....

The above sequence is an arithmetic sequence ( with a common difference of +1 ) i.e addition.

E.g :-

7 , 6 , 5 , 4 , 3 , 2 , 1 , .....

The above sequence is an arithmetic sequence ( with a common difference of -1 ) i.e subtraction.

The arithmatic sequence is categorized by a general formula. Which gives us the value of the term at (nth) position. The general formula for (nth) term in an arithmatic sequence is given by:

$\text{nth term value = a + (n-1)}\cdot d$

Where,

$\begin{gathered} a\colon\text{ The first term value in the sequence} \\ d\colon\text{ the common difference between successive terms} \\ n\colon\text{ The term number} \end{gathered}$

The questions pertains with an arithmatic sequence which is defined by the given formula:

$a_n\text{ = 3}\cdot n\text{ + 7}$

Where,

$\begin{gathered} a_n\colon\text{ The nth term value} \\ n\colon\text{ The term number} \end{gathered}$

We are to determine the ( 5th term ) in the sequence by using the formula already given in the question. So in other words:

$n=5,a_n\text{ = ?}$

To find the 5th term, we will simply plug in the value of ( n = 5 ) in the given arithmatic relation as follows:

$\begin{gathered} a_n\text{ = 3}\cdot(5)\text{ + 7} \\ a_n\text{ = 15 + 7} \\ a_n\text{ = 22 } \end{gathered}$

So the 5th term in the sequence would be:

$\textcolor{#FF7968}{22}$

We will go ahead and express the entire sequence:

n = 1 2 3 4 5 , ....

10 , 13 , 16 , 19 , 22 , ....

We can see the following two things:

$\begin{gathered} a\text{ = 10 ( first term value )} \\ d\text{ = 3 ( common difference )} \end{gathered}$

Now we will express using the general formula for nth term value:

$\begin{gathered} a_n\text{ = 10 + ( n - 1 ) }\cdot\text{ 3} \\ a_n\text{ = 10 + 3n - 3} \\ \textcolor{#FF7968}{a_n}\text{\textcolor{#FF7968}{ = 3n + 7}} \end{gathered}$

What we got above is the same formula given to us in the question. The only difference is that its a simplified version of the general formula used in arithmetic sequences.