Problem 23 - Algebra for College Students

Question

Find the sum of the first 75 terms of the sequence 5,1 , \(-3,-7, \ldots\) $$ -10,725 $$

Step-by-Step Solution

Answer

The sum of the first 75 terms is -10,725.

Step 1: Identify the Sequence Type

First, let's identify the type of sequence. In the given sequence 5, 1, -3, -7, we notice the difference between consecutive terms: 1 - 5 = -4, -3 - 1 = -4, -7 - (-3) = -4, which shows a constant difference. Hence, this is an arithmetic sequence with common difference \(d = -4\).

Step 2: Write the General Term Formula

The general term \(a_n\) of an arithmetic sequence is given by \(a_n = a_1 + (n-1) imes d\). Here, \(a_1 = 5\) and the common difference \(d = -4\). So, \(a_n = 5 + (n-1)(-4)\).

Step 3: Find the 75th Term

To find the 75th term \(a_{75}\), substitute \(n = 75\) in the formula \(a_n = 5 + (n-1)(-4)\):\[ a_{75} = 5 + (75-1)(-4) = 5 + 74(-4) = 5 - 296 = -291. \]

Step 4: Use the Sum Formula for Arithmetic Sequence

The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by: \(S_n = \frac{n}{2} (a_1 + a_n)\). We need to find \(S_{75}\), where \(a_1 = 5\) and \(a_{75} = -291\).

Step 5: Calculate the Sum of the First 75 Terms

Substitute \(n = 75\), \(a_1 = 5\), and \(a_{75} = -291\) in the sum formula \(S_n = \frac{n}{2} (a_1 + a_n)\): \[ S_{75} = \frac{75}{2} (5 + (-291)) = \frac{75}{2} \times (-286) = 75 \times (-143) = -10725. \]

Key Concepts

Sequence Type Identification

The first step in understanding sequences is to identify the type of sequence you are dealing with. In this problem, we start by examining the sequence: 5, 1, -3, -7, and so on. Look at the differences between consecutive terms. Here, the difference is consistently

• 1 - 5 = -4
• -3 - 1 = -4
• -7 - (-3) = -4

A constant difference between terms indicates an arithmetic sequence. Hence, this sequence is arithmetic with a common difference of \(d = -4\). Recognizing this sequence type helps apply specific formulas relevant to arithmetic sequences later.

General Term Formula

Once the sequence type is identified, constructing its general term formula is the next logical step. For any arithmetic sequence, the general term, denoted as \(a_n\), can be calculated using the formula:\[ a_n = a_1 + (n-1) \times d \]where

• \(a_1\) is the first term
• \(d\) is the common difference
• \(n\) is the term number in the sequence

For our sequence, \(a_1 = 5\) and \(d = -4\), so:\[ a_n = 5 + (n-1)(-4) \]This formula allows you to find any term's value by substituting the term's position into \(n\). It acts as a blueprint for exploring any individual term within the sequence.

Nth Term Calculation

To calculate a specific term in an arithmetic sequence, such as the 75th one, you’ll use the general term formula. Given the equation:\[ a_n = 5 + (n-1)(-4) \]we substitute \(n = 75\) to find \(a_{75}\):\[ a_{75} = 5 + (75-1)(-4) = 5 + 74(-4) \]\[ a_{75} = 5 - 296 = -291 \]Thus, the 75th term in the sequence is \(-291\). This calculation shows how the sequence progressively decreases as we proceed through its terms, and the general term formula is key in finding terms far along in the sequence.

Sum Formula for Arithmetic Sequences

Calculating the sum of terms in an arithmetic sequence involves using a specific sum formula. For the first \(n\) terms of an arithmetic sequence, the formula is:\[ S_n = \frac{n}{2} (a_1 + a_n) \]Where

• \(S_n\) is the sum of the first \(n\) terms
• \(a_1\) is the first term
• \(a_n\) is the \(n\)-th term

In this exercise, we need the sum of the first 75 terms:\(S_{75}\). With \(a_1 = 5\) and \(a_{75} = -291\), plug these values in:\[ S_{75} = \frac{75}{2} (5 + (-291)) \]\[ S_{75} = \frac{75}{2} \times (-286) = 75 \times (-143) \]\[ S_{75} = -10,725 \]This method provides a straightforward way to find the total of a large set of terms in an arithmetic sequence.