Question

In Exercises \(53-62,\) write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) \(1,4,7,10, \ldots\)

Short answer

The \(n\) th term of the sequence is given by the expression \(3n-2\)

Step-by-step solution

Identify the first term \(a_1\)

In the sequence, the first term \(a_1\) is 1.

Find the common difference 'd'

In the given sequence, the difference between consecutive terms is constant. This difference is 3 \(= 4 - 1 = 7 - 4 = 10 - 7\). Hence, the common difference 'd' is 3.

Use the general term of an arithmetic sequence

The general form of an arithmetic sequence is \(a_n = a_1 + (n-1)*d\). Substituting the values we have: \(a_1=1\), 'd'=3, we get \(a_n= 1+(n-1)*3 = 1+3n-3 = 3n-2\).

Final Answer

Therefore, the \(n\) th term of the sequence is given by the expression \(3n-2\)