Problem 39 - Algebra and Trigonometry

Question

39-44 Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=1, d=2, n=10$$

Step-by-Step Solution

Answer

The 10th partial sum of the sequence is 100.

Step 1: Understand the formula for the n-th partial sum

To find the partial sum of an arithmetic sequence, use the formula \( S_n = \frac{n}{2} (2a + (n-1)d) \), where \( n \) is the number of terms, \( a \) is the first term, and \( d \) is the common difference.

Step 2: Substitute known values

With \( a = 1 \), \( d = 2 \), and \( n = 10 \), substitute these into the partial sum formula: \[ S_{10} = \frac{10}{2} (2 imes 1 + (10-1) imes 2) \].

Step 3: Simplify the equation inside the brackets

Calculate the expression inside the brackets: \( 2 \times 1 = 2 \) and \( (10-1) \times 2 = 18 \). Thus, the equation becomes \( 2 + 18 = 20 \).

Step 4: Calculate the overall formula

Substitute the simplified value back into the formula: \[ S_{10} = \frac{10}{2} \times 20 \].

Step 5: Complete the calculation

Simplify \( \frac{10}{2} \) to get 5, and then multiply by 20: \( 5 \times 20 = 100 \). Thus, the partial sum \( S_{10} = 100 \).

Key Concepts

Partial Sum

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference. To find the partial sum of an arithmetic sequence, which refers to the sum of the first n terms of that sequence, we use a specific formula. This is the formula: \[ S_n = \frac{n}{2} (2a + (n-1)d) \] Here, \( S_n \) is the partial sum we want to find, \( a \) is the first term of the sequence, \( n \) is the number of terms we are summing, and \( d \) is the known common difference. Remember:

• Divide n by 2.
• Calculate 2 times the first term (\( 2a \)) and the product of (n-1) with the common difference (\( (n-1)d \)).
• Add these two results.

Take the result of these steps and multiply it by \( \frac{n}{2} \). This gives you the partial sum, a reliable way to quickly find out how much the first n numbers of the sequence add up to.

Common Difference

In an arithmetic sequence, there is something unchanging between every pair of numbers. This unchanging value is called the common difference. The common difference, denoted as \( d \), is what you add to each term to get to the next one. It's a simple yet powerful component because it maintains the uniformity of the sequence. Here's how it plays its part:

• Take any term in the sequence.
• Subtract the term immediately before it from this term.
• The result is the common difference, \( d \).

For instance, if you have a sequence starting with 1 and the next term is 3, subtract 1 from 3 to find that the common difference is 2. In any arithmetic sequence formula, like the formula for partial sums, the common difference helps calculate the increase required for each subsequent term.

N-th Term Formula

The n-th term of an arithmetic sequence is very straightforward to find once you understand the formula. Here is the formula you need: \[ a_n = a + (n-1)d \] In this expression, \( a_n \) is the n-th term you want to find, \( a \) is the first term of the sequence, \( n \) represents the position of the term in the sequence, and \( d \) is the common difference. You can follow these steps to find it:

• Begin with the first term \( a \).
• Multiply the common difference \( d \) by \( (n-1) \), which accounts for the number of times you need to add \( d \).
• Add this result to \( a \) to find the n-th term \( a_n \).

This formula is your key to unlocking any term within an arithmetic sequence directly, and it corresponds with and complements the formula for the sequence's partial sum.