Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

It The sum of the first 20 terms of an arithmetic sequence with a common difference of 3 is \(650 .\) Find the first term.

Step-by-Step Solution

Answer

The first term of the arithmetic sequence is \(4\).

Step 1: Understand the Problem

The problem is to find the first term of the arithmetic sequence. The sum of first 20 terms is given as \(650\), and a common difference, \(d\), is provided as \(3\). The formula for the sum of an arithmetic sequence, \(S_n = n/2 * (a_1 + a_n)\), will be used, where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.

Step 2: Apply the Formula

The arithmetic sequence formula \(S_n = n/2 * (a_1 + a_n)\) can be rewritten as \(a_n = (2S_n/n - a_1)\), which represents the \(n\)th term of the sequence. Based on the given data, the number of terms \(n\) is \(20\), \(S_n\) is \(650\), and \(d\) is \(3\). Plugging these into the formula, we get: \(a_n = a_1 + (n-1)*d\). Substituting the known values, we get: \(a_{20} = a_1 + (20-1)*3 = a_1 + 57\).

Step 3: Substitute the obtained a_n into the sum formula

Now, substitute the obtained \(a_{20}\) into the sum formula: \(S_{20} = 20/2 * (a_1 + a_{20})\). Hence, \(650 = 10*(a_1 + a_1 + 57)\). This gives a simplified equation: \(650 = 20*a_1 + 570\).

Step 4: Solve for the unknown

Solving the equation for \(a_1\): \(650 - 570 = 20*a_1\). This simplifies to \(80 = 20*a_1\). Finally, finding the value of \(a_1\) by dividing 80 by 20, we get \(a_1=4\).

Step 5: Conclusion

Submit the answer as the first term of this arithmetic sequence, which is \(4\).

Key Concepts

First Term

In an arithmetic sequence, the first term, denoted by \(a_1\), is the starting point of the sequence. It's incredibly important in determining all subsequent terms. An arithmetic sequence is a series of numbers in which each term increases by a constant amount known as the common difference.To find the first term when the sum of several terms is known, we use related formulas. In the given exercise, after applying these formulas, we solve the equation to determine the value of \(a_1\), which turns out to be 4. This means the sequence begins at 4, and subsequent terms build from this starting point. Understanding how to manipulate equations to find the first term is crucial in mastering arithmetic sequences.

Sum of Terms

The sum of terms in an arithmetic sequence can be found using the formula:\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]where:

• \(S_n\) is the sum of the first \(n\) terms,
• \(n\) is the number of terms to be added,
• \(a_1\) is the first term,
• and \(a_n\) is the last term.

This formula helps in finding the total when you know the number of terms, the first term, and the last term. In the original exercise, the sum of the first 20 terms is given as 650, which is used to find the unknown first term. This simplicity in formula showcases the elegant consistency of arithmetic sequences.

Common Difference

The common difference, often represented as \(d\), is the consistent amount that each term in an arithmetic sequence increases or decreases by. It's calculated by subtracting any term from the subsequent term.In the problem, the common difference is 3. This means each term is 3 more than the one before. Recognizing the common difference is essential for predicting future terms in the sequence or confirming that a sequence is arithmetic.Here’s how the common difference affects a sequence:

• Starting from the first term, each subsequent term is calculated as \(a_1 + d\), \(a_1 + 2d\), and so on.
• For the 20th term, the formula \(a_{20} = a_1 + 19d\) is used, showing the accumulated impact of the common difference across the sequence.

Sequence Formula

The sequence formula for arithmetic sequences enables us to find any term in the sequence or even the sum of terms up to a certain point. It’s expressed as:\[ a_n = a_1 + (n-1)d \]where:

• \(a_n\) indicates the \(n\)th term,
• \(a_1\) is the first term,
• \(d\) is the common difference,
• \(n\) indicates the specific term number we want to find.

Using the sequence formula, we find the 20th term of the sequence as part of solving the exercise. The ability to apply this straightforward formula is a key skill for working with arithmetic sequences, providing a direct route to understanding or computing any part of the sequence.