Problem 37 - Algebra and Trigonometry

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: Understand the Problem

We need to find the first term of an arithmetic sequence, given that the 10th term is \(\frac{55}{2}\) and the 2nd term is \(\frac{7}{2}\). An arithmetic sequence follows the formula for the nth term: \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term and \(d\) is the common difference.

Step 2: Equation for Second Term

Using the formula for the nth term \(a_n = a_1 + (n-1) \cdot d\), set up the equation for the second term: \(a_2 = a_1 + (2-1) \cdot d = a_1 + d\). We know that \(a_2 = \frac{7}{2}\), so we have \(a_1 + d = \frac{7}{2}\).

Step 3: Equation for Tenth Term

Similarly, set up the equation for the 10th term: \(a_{10} = a_1 + (10-1) \cdot d = a_1 + 9d\). We know that \(a_{10} = \frac{55}{2}\), so we have \(a_1 + 9d = \frac{55}{2}\).

Step 4: Solve for the Common Difference

We have two equations: \(a_1 + d = \frac{7}{2}\) and \(a_1 + 9d = \frac{55}{2}\). Subtract the first equation from the second equation to eliminate \(a_1\): \[(a_1 + 9d) - (a_1 + d) = \frac{55}{2} - \frac{7}{2} \Rightarrow 8d = \frac{48}{2} \Rightarrow 8d = 24\]. Solve for \(d\) by dividing both sides by 8: \(d = 3\).

Step 5: Solve for the First Term

Substitute \(d = 3\) back into the equation \(a_1 + d = \frac{7}{2}\): \[a_1 + 3 = \frac{7}{2}\]. Subtract 3 from both sides to solve for \(a_1\): \[a_1 = \frac{7}{2} - \frac{6}{2} \Rightarrow a_1 = \frac{1}{2}\].

Key Concepts

Common Difference

An arithmetic sequence is defined by its characteristic feature, the common difference. This term refers to the constant gap, or difference, between consecutive elements in the sequence. It's one of the most important parts of an arithmetic sequence as it dictates the rate at which the sequence progresses, much like steps climbing a staircase.
To find the common difference, we employ the formula for the nth term, given by:

• \( a_n = a_1 + (n-1) \, d \)

Here, \( d \) represents the common difference. From this formula, subtracting any two consecutive terms gives \(d\). In our original problem, by solving the system of equations and using terms 2 and 10, we arrived at a solution where \( d = 3 \). This demonstrates how solving simultaneous equations of different terms in the given sequence can reveal the common difference, a crucial concept in understanding sequence movement.

Nth Term

Finding the nth term of an arithmetic sequence allows one to determine any term within the sequence without listing all preceding terms. This is incredibly useful for sequences with a large number of terms.
The formula for calculating the nth term is:

• \( a_n = a_1 + (n-1) \, d \)

This formula states that the nth term, \( a_n \), is equal to the first term \( a_1 \) plus the result of multiplying the common difference \( d \) by one less than the position number of the term \(n-1\). This approach was used in our exercise to determine values for different terms, allowing us to set up equations based on known terms like the second and tenth. Understanding this formula is key because it transforms the sequence's general structure into specific terms, making it easier to solve for unknowns.

First Term

The first term of an arithmetic sequence, denoted \( a_1 \), is the starting point of the sequence. It acts as the reference point from which all other terms can be derived through the pattern dictated by the common difference.
To find the first term when not given directly, the nth-term formula is again utilized, as it incorporates \( a_1 \) as a part of the expression:

• \( a_n = a_1 + (n-1) \, d \)

In our task, knowing the values of specific terms, like the second and tenth terms, helped establish equations using this formula. By systematically eliminating variables, we can solve for \( a_1 \). In this case, after determining \( d = 3 \), substitution back into the equation for the second term revealed \( a_1 = \frac{1}{2} \). The method highlights the interdependence of arithmetic sequence components and showcases the logical approach needed to uncover the initial term without explicit information.