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Worked solution

Count terms using the last-term condition

Find the number of terms in the arithmetic progression 57,54,51,…,357,54,51,\ldots,3.

CJC · Sequences and Series · Example 7

Answer

19 terms.

Compare your reasoning

Identify the data

a=57,d=−3a=57,\qquad d=-3

The progression decreases by three.

Set the nth term equal to the final term

57+(n−1)(−3)=3,60−3n=3,3n=57,n=19.\begin{aligned}57+(n-1)(-3)&=3,\\60-3n&=3,\\3n&=57,\\n&=19.\end{aligned}

The complete source chain retains the positive integer index.

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Common mistakes

Counting differences alone gives eighteen gaps; the progression contains nineteen terms.