Problem 17 - Algebra and Trigonometry

Question

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{s 0}\) when \(a_{1}=7, d=5\)

Step-by-Step Solution

Answer

The 50th term of the arithmetic sequence is 252.

Step 1: Identifying Given Values

In this problem, we are given the first term (\(a_{1}\)) of the arithmetic sequence as 7 and the common difference (d) as 5. We are asked to find the 50th term (\(a_{50}\)).

Step 2: Apply the Formula for the nth term of an Arithmetic Sequence

The formula for the nth term of an arithmetic sequence is \(a_{n} = a_{1} + (n-1)*d\). We can substitute the values into the formula: \(a_{50} = 7 + (50 - 1)*5\).

Step 3: Solve the Equation

Now, we simplify the equation: \(a_{50} = 7 + 49*5\), then \(a_{50} = 7 + 245\) and finally \(a_{50} = 252\).

Key Concepts

Arithmetic Progression

An arithmetic progression, or AP for short, is one of the simplest forms of sequences in mathematics. It is a sequence where each term after the first is found by adding a constant value, known as the common difference, to the previous term. Imagine you're climbing stairs, and with each step, you go up the same height; that height is like the common difference in an arithmetic sequence.

Consider the scenario of stepping onto a bus with an initial step height (the first term) and each subsequent step (term) being of equal height (common difference). The sequence of steps creates a pattern that can be easily predicted once you know the height of that first step and the height each subsequent step adds. This predictability is what makes arithmetic progressions valuable in various fields like finance, computer science, and physics, where regular intervals are a common occurrence.

Sequence and Series

Sequences and series are fundamental concepts in mathematics that deal with ordered lists of numbers and their sum, respectively. A sequence is like a treasure map, giving us directions as to where we'll find each treasure (or term) one after another. The term 'arithmetic sequence' refers to a sequence where the difference between consecutive terms is always the same.

In contrast, if you were to add up the terms of a sequence, you would have a series. It is like collecting all the treasures you've found along your journey and counting them all together. Sequences are about the individual steps or elements, while series focus on the cumulative total. This distinction is crucial when analyzing patterns and solving problems in various mathematical contexts.

Common Difference

The common difference is the consistent interval between consecutive terms in an arithmetic sequence. It is what defines the uniqueness of the sequence. This constant step can be positive, negative, or even zero; a positive common difference means the sequence increments, while a negative one implies it decrements. A zero difference would result in a constant sequence, where all terms are equal.

Understanding the common difference is like knowing the rhythm in music. Just as a steady beat guides a song, the common difference guides the growth or decline of an arithmetic progression. The concept of the common difference not only helps predict future terms but also provides insight into the nature of the sequence's trend. Thus, grasping the common difference is vital for students exploring arithmetic progression and seeking to solve related problems.