Problem 68 - College Algebra

Question

Explain how to write terms of a sequence if the formula for the general term is given.

Step-by-Step Solution

Answer

To write the terms of a sequence using the general term formula, substitute the term numbers 1, 2, 3, etc. into the formula, calculating each term individually. For \(\(n=1\), the first term equals the starting term. For subsequent terms, increase \(n\) by 1 each time and execute the calculation according to the formula.

Step 1: Understand the Formula

In an arithmetic sequence, each term can be expressed in terms of the first term \(a\), common difference \(d\), and the term number \(n\). The formula is \(a_n = a + (n-1)d\).

Step 2: Substituting for n

To get the individual terms of the sequence, substitute the term number for \(n\) in the formula. For example, to get the first term (\(a_1\)), substitute \(n=1\) into the formula \((a_1 = a + (1-1)d = a)\). As expected, the first term, \(a_1\), equals to the starting term \(a\).

Step 3: Get the Subsequent Terms

For the second term, substitute \(n=2\) into the formula \((a_2 = a + (2-1)d)\), and so on. Continue substituting the values \(3,4,5, \ldots\) for \(n\) to get the 3rd, 4th, 5th, etc. terms of the sequence.