Problem 48 - Algebra and Trigonometry

Question

45–50 A partial sum of an arithmetic sequence is given. Find the sum. $$-10-9.9-9.8-\dots-0.1$$

Step-by-Step Solution

Answer

The sum of the sequence is -505.

Step 1: Identify the First Term and Last Term

In the arithmetic sequence, the first term \( a_1 \) is \(-10\) and the last term \( a_n \) is \(-0.1\).

Step 2: Determine the Common Difference

To find the common difference \( d \), subtract the first term from the second term: \(-9.9 - (-10) = 0.1\). Hence, the common difference is \( d = 0.1 \).

Step 3: Find the Number of Terms

The formula for the nth term \( a_n \) of an arithmetic sequence is \( a_n = a_1 + (n-1)d \). We have \( a_n = -0.1 \), \( a_1 = -10 \), and \( d = 0.1 \). Plug these into the formula to find \( n \):\[-0.1 = -10 + (n-1) \times 0.1\] \[9.9 = (n-1) \times 0.1\] \[(n-1) = \frac{9.9}{0.1} = 99\] Thus, the number of terms \( n = 100 \).

Step 4: Use the Formula to Find the Sum of the Sequence

The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is:\[S_n = \frac{n}{2} \times (a_1 + a_n)\]Substitute the known values of \( n \), \( a_1 \), and \( a_n \) into the formula:\[S_{100} = \frac{100}{2} \times (-10 + (-0.1)) = 50 \times (-10.1) = -505\]

Step 5: Verify the Calculation

Recalculate the sum using the step-by-step method to ensure the arithmetic is correct. This confirms that the sum is \( -505 \).

Key Concepts

Sum of Arithmetic Sequence

In mathematics, the sum of an arithmetic sequence is the total of all terms added together. Arithmetic sequences have a key characteristic—each successive term is equal to the sum of the previous term and a constant value, called the common difference. To find the sum of an arithmetic sequence, you can use the formula:

• \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
• Here, \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the nth (or last) term.

This formula effectively finds the average of the first and last terms and multiplies it by the number of terms. It's a very efficient way to calculate the sum without adding every individual term manually. In this example, the first term is \(-10\), the last term is \(-0.1\), and the number of terms is 100. Substituting these into the formula gives:\[ S_{100} = \frac{100}{2} \times (-10 + (-0.1)) = 50 \times (-10.1) = -505 \]Hence, the sum of the sequence is \(-505\).

Common Difference

The common difference in an arithmetic sequence is the consistent interval between consecutive terms. It is what makes the sequence 'arithmetic'. To find this common difference, you subtract the first term from the second term. This subtraction gives the value by which the sequence increases or decreases.For example, in the sequence \(-10, -9.9, -9.8, \ldots\), the first term \( a_1 \) is \(-10\) and the second term is \(-9.9\). To find the common difference \( d \), you calculate:

• \( d = -9.9 - (-10) = 0.1 \)

So, every term in the sequence is just 0.1 more than the previous one. Understanding the common difference is crucial, as it's used to determine other properties of the sequence, including the nth term and the entire sequence's sum.

Number of Terms

Determining the number of terms in an arithmetic sequence is essential to solve many related problems, including finding the sum. For sequences where the first term \( a_1 \), the last term \( a_n \), and the common difference \( d \) are known, you can find the number of terms \( n \) using the formula for the nth term:

• \[ a_n = a_1 + (n-1) \times d \]

Plug the known values into the formula to solve for \( n \). In our problem, we have the last term \(-0.1\), the first term \(-10\), and the common difference \(0.1\):

• \[ -0.1 = -10 + (n-1) \times 0.1 \]
• Solving this gives:\[ 9.9 = (n-1) \times 0.1 \]
• \[ n-1 = \frac{9.9}{0.1} = 99 \]
• \( n = 100 \)

So, there are 100 terms in the sequence.

Nth Term Formula

The nth term formula of an arithmetic sequence allows you to find any term in the sequence without listing all the preceding terms. If you know the first term and the common difference, you can easily find the nth term using this pivotal formula:

• \[ a_n = a_1 + (n-1) \times d \]

Here, \( a_n \) is the nth term you're trying to find, \( a_1 \) is the first term, and \( d \) is the common difference between terms. This formula helps in determining individual terms of the sequence or even cross-verifying results.For instance, if you need to confirm a specific term in our arithmetic sequence starting from \(-10\) with a common difference of 0.1, this formula comes handy. It is crucial for both theoretical understanding and practical problem solving in arithmetic progressions.