Problem 70 - Algebra and Trigonometry

Question

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \\{2 n-5\\} $$

Step-by-Step Solution

Answer

Arithmetic sequence, common difference is 2, sum of the first 50 terms is 2300.

Step 1 - Identify the Sequence Type

Examine the given sequence formula \(2n - 5\). Check for a common difference (arithmetic) or a common ratio (geometric) by plugging in different values of \(n\).

Step 2 - Compute Initial Terms

Calculate the first few terms by substituting various values of \(n\): \(n=1\) gives \(2(1)-5=-3\), \(n=2\) gives \(2(2)-5=-1\), \(n=3\) gives \(2(3)-5=1\).

Step 3 - Check Differences

Compute the differences between consecutive terms: \(-1 - (-3) = 2\), and \(1 - (-1) = 2\). The differences are constant, indicating an arithmetic sequence.

Step 4 - Determine the Common Difference

Since the differences between terms are constant, the common difference \(d\) is \(2\).

Step 5 - Verify Arithmetic Sequence

An arithmetic sequence can be written as \(a_n = a + (n-1)d\). In this problem, \(a_1 = -3\) and \(d = 2\), so \(a_n = -3 + (n-1) * 2 = 2n - 5\).

Step 6 - Find the Sum of the First 50 Terms

The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by \(S_n = \frac{n}{2} (2a + (n-1)d)\). Substituting \(a = -3\), \(d = 2\), and \(n = 50\): \[S_{50} = \frac{50}{2} (2(-3) + (50-1)2)\] Simplifying, \[S_{50} = 25(-6 + 98) = 25 * 92 = 2300\].

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the 'common difference'. In the given problem, we have a formula for the sequence as \(2n - 5\). To identify if it's an arithmetic sequence, we calculate whether the difference between each term remains the same. By plugging in different values of \(n\) (such as 1, 2, 3), we got the terms \(-3, -1, 1\). The differences between these terms are all equal to 2, confirming it's an arithmetic sequence.

Common Difference

The common difference (denoted as \(d\)) in an arithmetic sequence is found by subtracting any term from the term that follows it.
In the sequence given by \(2n - 5\), as we calculated, the first few terms are \(-3, -1, 1\).
By subtracting consecutive terms:

• \(-1 - (-3) = 2\)
• \(1 - (-1) = 2\)

We see the difference is consistently 2. So, the common difference for this sequence is 2.

Sum of Arithmetic Series

The sum of the first \(n\) terms of an arithmetic sequence can be calculated using the formula: ewline \(S_n = \frac{n}{2} (2a + (n-1)d)\) where:

• \(S_n\) is the sum of the first \(n\) terms
• \(a\) is the first term
• \(d\) is the common difference
• \(n\) is the number of terms

For the given sequence, \(2n - 5\), we calculated \(a = -3\) and \(d = 2\), and we need to find the sum of the first 50 terms. ewline Using the formula: \(\begin{aligned} S_{50} &= \frac{50}{2} (2(-3) + (50-1) \times 2) \ &= 25(-6 + 98) \ &= 25 \times 92 \ &= 2300 \end{aligned}\)ewlineSo, the sum of the first 50 terms of this arithmetic sequence is \(2300\).