Question

1. error analysis describe and correct the error in finding the common difference of the arithmetic sequence. 2, 1, 0, -1,... each term is 1 less than the previous term. so, the common difference is 1.

Explanation:

Step1: Recall common difference formula

The common difference \( d \) of an arithmetic sequence is given by \( d = a_{n + 1}-a_{n} \), where \( a_{n+1} \) is the next term and \( a_n \) is the current term.

Step2: Analyze the error

In the sequence \( 2, 1, 0, - 1,\dots \), to find the common difference, we calculate \( 1 - 2=-1 \), \( 0 - 1=-1 \), \( - 1-0 = - 1 \). The error is in the conclusion of the common difference value. Even though each term is 1 less than the previous term, the common difference (which is \( a_{n + 1}-a_{n} \)) should be \(- 1\), not 1. The mistake was in the sign of the common difference.

Step3: Correct the common difference

Using the formula \( d=a_{n + 1}-a_{n} \), for \( n = 1 \), \( a_1 = 2 \), \( a_2=1 \), so \( d=1 - 2=-1 \). Checking with other terms: \( 0 - 1=-1 \), \( - 1-0=-1 \). So the common difference is \(-1\).

Answer:

The error is that the common difference was incorrectly stated as 1; the correct common difference is \(-1\) (since \( a_{n + 1}-a_{n}=-1 \) for consecutive terms in the sequence \( 2,1,0,-1,\dots \)).