Problem 20

Question

For the following exercises, fi d the fi st term given two terms from an arithmetic sequence. Find the fi st term or $a_{1}$ of an arithmetic sequence if $a_{7}=21$ and $a_{15}=42$.

Step-by-Step Solution

Verified

Answer

The first term of the sequence is $ 5.25 $.

1Step 1: Understanding the Arithmetic Sequence Formula

An arithmetic sequence is defined by 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.

2Step 2: Write Equations for Given Terms

Using the formula $ a_n = a_1 + (n-1) \cdot d $, we can write two equations based on the given terms. For $ a_7 = 21 $, the equation is $ a_1 + 6d = 21 $. For $ a_{15} = 42 $, the equation is $ a_1 + 14d = 42 $.

3Step 3: Subtract Equations to Solve for d

Subtract the first equation from the second to eliminate $ a_1 $ and solve for $ d $: $ (a_1 + 14d) - (a_1 + 6d) = 42 - 21 $. This simplifies to $ 8d = 21 $, so $ d = \frac{21}{8} $.

4Step 4: Solve for the First Term a1

Substitute $ d = \frac{21}{8} $ back into the first equation: $ a_1 + 6 \cdot \frac{21}{8} = 21 $. This becomes $ a_1 + \frac{126}{8} = 21 $. Simplify $ \frac{126}{8} $ to $ 15.75 $, then solve for $ a_1 $: $ a_1 = 21 - 15.75 = 5.25 $.

5Step 5: Verify Solution with Second Equation

Verify $ a_1 = 5.25 $ using the second equation: $ a_1 + 14d = 42 $. Substitute $ d = \frac{21}{8} $, we get $ 5.25 + 14 \cdot \frac{21}{8} = 42 $. Simplify and check to confirm it is accurate.

Key Concepts

First Term CalculationCommon Difference in SequencesSolve Sequence Equations

First Term Calculation

Many students find calculating the first term of an arithmetic sequence a bit tricky at first, mostly because it involves a couple of steps. But once you understand the way sequences work, it becomes much easier! In every arithmetic sequence, all terms are generated using a specific formula:

$ a_n = a_1 + (n-1) \cdot d $
Where $ a_1 $ is the first term you want to find
$ d $ is the common difference between terms

For example, to find the first term when you know two other terms in the sequence, you can set up equations based on the known terms and their positions, then solve for $ a_1 $. This method helps uncover the unknown starting point of your sequence. It's similar to solving a puzzle, where you align all pieces to reveal the whole picture. Remember, every sequence starts at its first term, so focus on unlocking that $ a_1 $.

Common Difference in Sequences

The common difference, denoted as $d$, is key to understanding how an arithmetic sequence progresses from one term to the next. In simple terms, it's the number you add or subtract consistently to move from one term to the next in your sequence. To find this number, you need information from two known terms in the sequence. Once again, our trusty arithmetic sequence formula comes into play.

Use: $ a_n = a_1 + (n-1) \cdot d $
Write equations based on the known sequence terms
Subtract these equations to solve for $ d $

This process allows you to determine the incremental step or gap between terms, providing a clearer understanding of the sequence's structure. For example, in our exercise, we found that $d = \frac{21}{8}$, meaning there's a modest fractional increase between each term.

Solve Sequence Equations

Solving sequence equations might sound complicated at first, but breaking the problem into parts can simplify the process. You usually start by forming equations with the known terms of the sequence. For instance, given terms $a_7 = 21$ and $a_{15} = 42$, you can write two separate equations using the arithmetic sequence formula. After setting these equations, use them to find the common difference $d$. For instance:

Equation 1: $ a_1 + 6d = 21 $
Equation 2: $ a_1 + 14d = 42 $

By subtracting these equations, we remove $a_1$ and find $d$, and with $d$ known, substitute back to discover the first term. Eventually, verify your solution with the other equation to ensure every piece fits perfectly. Always check the accuracy of your results within the context of the sequence to maintain the integrity of your solution.

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