Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a formula for \(a_{n}\) for the arithmetic sequence. $$a_{1}=15, d=4$$

Step-by-Step Solution

Answer

The formula for the nth term of the arithmetic sequence is \(a_{n} = 4n + 11\).

Step 1: Understand the formula for an arithmetic sequence

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant is commonly known as the common difference, represented as \(d\). The formula for finding the nth term, \(a_{n}\), of an arithmetic sequence is given by \(a_{n}=a_{1}+(n-1)d\), where \(a_{1}\) is the first term and \(d\) is the common difference.

Step 2: Substitute the given values into the formula

Next, use the given values for the first term, \(a_{1}=15\), and the common difference, \(d=4\), in the formula. This results in \(a_{n} = 15 + (n-1)4\).

Step 3: Simplify the formula

The last step is to simplify the formula to find \(a_{n}\). Distribute the 4 in the expression \(4(n-1)\) to get \(4n - 4\). Then, add 15 to obtain the final answer: \(a_{n} = 4n + 11\).