Chapter 4, Problem 16CT

To determine

To find:Whether the sequence is arithmetic. If the sequence is arithmetic, write a recursive formula and an explicit formula to represent it.

Answer to Problem 16CT

The sequence is arithmetic.

The recursive formula is $A(n)=A(n-1)+0.25$ , where $A(1)=3$ .

The explicit formula is $A(n)=3+(n-1)0.25$ .

Explanation of Solution

Given information: $3,3.25,3.5,3.75,\mathrm{...}$

Calculation:

Step-1:

Difference between 2nd term and 1st term $=3.25-3=0.25$

Difference between 3rd term and 2nd term $=3.5-3.25=0.25$

Difference between 4th term and 3rd term $=3.75-3.5=0.25$

Since, the sequence has a common difference of 0.25 the sequence is arithmetic.

Step-2:

Let $n=$ the term number in the sequence.

Let $A(n)=$ the value of the nth term of the sequence.

Value of term 1 $=A(1)=3$

Value of term 2 $=A(2)=A(1)+0.25=3+0.25=3.25$

Value of term 3 $=A(3)=A(2)+0.25=3.25+0.25=3.5$

Value of term 4 $=A(4)=A(3)+0.25=3.5+0.25=3.75$

Value of term n $=A(n)=A(n-1)+0.25$

The recursive formula for the arithmetic sequence is $A(n)=A(n-1)+0.25$ , where $A(1)=3$ .

Step-3

The general form of an explicit formula is,

  $A(n)=A(1)+(n-1)d$

Substituting $A(1)=3$ and $d=0.25$ ,

  $\begin{array}{l}A(n)=A(1)+(n-1)d\\ \Rightarrow A(n)=3+(n-1)0.25\end{array}$

The explicit formula for the arithmetic sequence is $A(n)=3+(n-1)0.25$ .