Problem 44 - Algebra 2

Question

Use the values of \(a_{1}\) and \(S_{n}\) to find the value of \(a_{n}\) $$ a_{1}=-6 \text { and } S_{50}=-5150 ; a_{50} $$

Step-by-Step Solution

Answer

The 50th term of the sequence, \(a_{50}\), is 86.

Step 1: Find the Common Difference

Calculate the common difference \(d\) with the formula \(d = \frac{{2S_{n} - na_{1}}}{{n(n-1)/2}}\), the difference between any two consecutive terms. Substituting \(S_{50} = -5150\), \(a_{1} = -6\), and \(n = 50\) into the formula, we get \(d = \frac{{2(-5150) - 50(-6)}}{{50(50-1)/2}} = 2\).

Step 2: Compute \(a_{50}\)

Having obtained \(d\), we can calculate \(a_{n}\) using the formula \(a_{n} = a_{1} + (n - 1)d\). Substituting \(a_{1} = -6\), \(d = 2\), and \(n = 50\) into the formula, we get \( a_{50} = -6 + (50 - 1)2 = 86\)

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is an important characteristic. Simply put, it is the difference between any two consecutive terms in the sequence. To find the common difference, you usually subtract the previous term from the current term. In this particular exercise, the common difference can also be calculated using a more sophisticated formula: \(d = \frac{{2S_{n} - na_{1}}}{{n(n-1)/2}} \). Here's how it works:

• The symbol \(S_n\) represents the sum of the first \(n\) terms.
• \(a_1\) is the first term, and \(n\) is the number of terms.
• Plugging these values into the formula will give you \(d\), the common difference.

In the original solution, applying these values gave a common difference \(d = 2\). Knowing this difference allows us to determine subsequent terms in the sequence easily.

Sum of n terms

The sum of the first \(n\) terms of an arithmetic sequence is denoted by \(S_n\). This sum can be calculated using a specific formula: \[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \] where:

• \(n\) is the number of terms you want to sum.
• \(a_1\) is the first term.
• \(d\) is the common difference.

This formula helps you find out the total of several terms without listing all of them explicitly. In our exercise, we knew \(S_{50} = -5150\). This was crucial for finding the common difference with another formula, and subsequently, the value of \(a_{50}\). Being able to calculate \(S_n\) means you can handle large arithmetic sequences effortlessly.

Nth Term Formula

The nth term formula in an arithmetic sequence allows us to find any specific term without listing out the whole sequence. This formula is given by: \[ a_n = a_1 + (n - 1)d \] Here is what each symbol signifies:

• \(a_n\) is the term you wish to find.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference between the terms.
• \(n\) represents the position of the term in the sequence.

In our exercise, this formula was used to find \(a_{50}\). Substituting our known values—\(a_1 = -6\), \(d = 2\), and \(n = 50\)—into the formula, it resulted in \(a_{50} = 86\). This shows how powerful the formula is, as it provides a quick method to find terms in large sequences.