Chapter 9.1, Problem 6LC

Solution

To determine the student’s error and write a correct explicit formula.

  $a_n=3n-2$

Given:

The given sequence is:

  $1,4,7,10,\dots$

Calculation:

The student writes an explicit formula for given sequence as:

  $a_n=3n+1$

An explicit formula is a formula that describes the $n^{th}$ term of a sequence using the number $n.$ To find the explicit formula, the first step is to identify the pattern between two constative terms and then find the explicit formula.

Now, for the given sequence, the difference between first two terms is $4-1=3$ .And the difference between second and third term is $7-4=3$ . And the difference between the third and fourth term is again $10-7=3$ and so on. So, the term $3n$ is involved in the given sequence.

Now, to find the given sequence, subtract $2$ from each term of $3n$ to create an explicit formula as:

  $a_n=3n-2$

where

  $n=\text{ position of the unknown term}$

Here, the student’s error is the wrong explicit formula for the given sequence, as the value of $n$ should start from $1$ . The explicit formula considered by the student assumes that the value of $n$ will start from $0$ .