Problem 37 - Precalculus Mathematics for Calculus

Question

The tenth term of an arithmetic sequence is \(\frac{55}{2},\) and the second term is \(\frac{7}{2} .\) Find the first term.

Step-by-Step Solution

Answer

The first term is \(\frac{1}{2}\).

Step 1: Identify the Formula for the nth Term

The nth term of an arithmetic sequence is given by the formula \(a_n = a + (n-1)\cdot d\), where \(a\) is the first term and \(d\) is the common difference.

Step 2: Set Up Equations for Known Terms

We know that the second term is \(a_2 = \frac{7}{2}\) and the tenth term is \(a_{10} = \frac{55}{2}\). Using the formula for the nth term: - For the second term: \(a + d = \frac{7}{2}\)- For the tenth term: \(a + 9d = \frac{55}{2}\).

Step 3: Solve the System of Equations

Subtract the first equation from the second equation: \[(a + 9d) - (a + d) = \frac{55}{2} - \frac{7}{2}\]This simplifies to: \[8d = 24\]Divide both sides by 8 to solve for \(d\).\[d = 3\]

Step 4: Find the First Term using d

Substitute \(d = 3\) back into the equation \(a + d = \frac{7}{2}\) to find \(a\):\[a + 3 = \frac{7}{2}\]\[a = \frac{7}{2} - 3 = \frac{7}{2} - \frac{6}{2} = \frac{1}{2}\]

Step 5: Verify the Solution

With \(a = \frac{1}{2}\) and \(d = 3\), verify the tenth term:\[a_{10} = a + 9d = \frac{1}{2} + 9 \cdot 3 = \frac{1}{2} + \frac{27}{2} = \frac{55}{2}\].This confirms our solution is correct.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial component. This term refers to the consistent interval between consecutive terms in a sequence. In simpler words, it's what you add (or subtract) every time to get from one term in the sequence to the next. Each term increases by the same value if the sequence is ascending, or decreases if it's descending.

To find the common difference, you can subtract any term from the subsequent term in the sequence. In our exercise, we had the equations for the second and tenth terms, as follows:

• The second term equation: \( a + d = \frac{7}{2} \)
• The tenth term equation: \( a + 9d = \frac{55}{2} \)

By setting up a simple subtraction of these two equations, we could isolate the common difference. This approach allowed us to determine that the common difference \(d\) is 3.

Knowing the common difference is essential as it helps in predicting the subsequent terms of the sequence.

First Term

The first term in an arithmetic sequence is where it all begins. This term is often denoted by \(a\) and serves as the foundation from which the entire sequence is built. Understanding or determining the first term is advantageous, as it allows you to determine the entire sequence when combined with the common difference.

In solving problems like our exercise, once you know the common difference and have an equation involving it, you can substitute back into any equation to find the first term. In this instance, we substituted the common difference \(d = 3\) into the equation for the second term, \(a + d = \frac{7}{2}\), to find \(a\).

This substitution confirmed that the first term \(a\) of our sequence is \( \frac{1}{2} \). Knowing this starting point is vital as it sets the stage for identifying every subsequent term in the sequence.

Nth Term Formula

The nth term formula in arithmetic sequences is a tool that unravels the mystery of the whole sequence. This formula provides a way to calculate any term in the sequence without needing to manually add the common difference repeatedly. The formula is structured as:\[a_n = a + (n-1)\cdot d\]Where:

• \(a_n\) is the term you want to find
• \(a\) is the first term
• \(d\) is the common difference
• \(n\) is the position of the term in the sequence

This formula is highly effective in quickly determining any term's value. In the context of our exercise, it enabled us to set up equations for both the second and tenth terms using the known values of \(a\) and \(d\).

By understanding this formula, you empower yourself to not only predict sequence terms but also to solve complex problems that involve sequences. Thus, mastering the nth term formula is central to mastering arithmetic sequences.