Question

What is the 6th term in the sequence? f(1) = 10 f(n) = f(n-1) + 3

Answer 1

We will investigate the evaluation of recursive sequences.

A iterative sequences are categorized by two values i.e one value preceding and the current value. The current value is evaluated on the basis of the previous value. In other words the current value is a function of preceeding value or depends on it.

All iterative relations of a sequence are expressed mathematically by two values as follows:

`\begin{gathered} f\text{ ( n ): Current nth term value} \\ f\text{ ( n - 1 ): preceeding/ previous value} \end{gathered}`

A iterative relation is a unique mathematical relationship between the current term f ( n ) and the previous value given as follows:

`\textcolor{#FF7968}{f}\text{\textcolor{#FF7968}{ ( n ) = f ( n - 1 ) + 3}}`

For this particular relationship the first term value is given as follows:

`\textcolor{#FF7968}{n}\text{\textcolor{#FF7968}{ = 1, f ( 1 ) = 10}}`

We will now start determining the terms in the sequence by plugging the respective term numbers ( n ) and use the mathematical relationship for the sequence as follows:

`\begin{gathered} \text{\textcolor{#FF7968}{For n = 2,}} \\ f\text{ ( 2 ) = f ( 1 ) + 3} \\ f\text{ ( 2 ) = 10 + 3} \\ \textcolor{#FF7968}{f}\text{\textcolor{#FF7968}{ ( 2 ) = 13}} \end{gathered}`

We have evaluated the 2nd term in the sequence as 13. We will repeat the above steps for all term numbers ( n = 3 , 4 , 5 , 6 ).

`\begin{gathered} \text{\textcolor{#FF7968}{For n = 3,}} \\ f\text{ ( 3 ) = f ( 2 ) + 3} \\ f\text{ ( 3 ) = 13 + 3} \\ \textcolor{#FF7968}{f}\text{\textcolor{#FF7968}{ ( 3 ) = 16}} \\ \\ \text{\textcolor{#FF7968}{For n = 4,}} \\ f\text{ ( 4 ) = f ( 3 ) + 3} \\ \text{f ( 4 ) = 16 + 3} \\ \text{\textcolor{#FF7968}{f ( 4 ) = 19 }} \\ \\ \text{\textcolor{#FF7968}{For n = 5,}} \\ f\text{ ( 5 ) = f ( 4 ) + 3} \\ f\text{ ( 5 ) = 19 + 3} \\ \textcolor{#FF7968}{f}\text{\textcolor{#FF7968}{ ( 5 ) = 22}} \end{gathered}`

And them lastly for the 6th term in the sequence as follows:

`\begin{gathered} \text{\textcolor{#FF7968}{For n = 6,}} \\ f\text{ ( 6 ) = f ( 5 ) + 3} \\ f\text{ ( 6 ) = 22 + 3} \\ \text{\textcolor{#FF7968}{f ( 6 ) = 25}} \end{gathered}`

Hence, the answer to the 6th term number is:

`\textcolor{#FF7968}{25}`