Problem 47 - Algebra and Trigonometry

Question

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. $$ \sum_{i=1}^{30}(-3 i+5) $$

Step-by-Step Solution

Answer

The first three terms of the sequence are 2, -1 and -4. The last term is -85. The sum of all 30 terms is -1245.

Step 1 – Generate the first three terms and the last term

To find the terms, simply substitute the index i with the term number in the formula -3i + 5. The first term \(a_1 = -3(1) + 5 = 2\), the second term \(a_2 = -3(2) + 5 = -1\), the third term \(a_3 = -3(3) + 5 = -4\) and finally the last term which is the 30th, \(a_{30} = -3(30) + 5 = -85\).

Step 2 – Apply the sum formula

The sum \(S_n\) of the first n terms of an arithmetic sequence can be calculated using 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 difference between the terms, which in this case is -3 since this is the coefficient of i. Thus, we can plug in the values: \(S_{30} = \frac{30}{2} [2 * 2 + (30 - 1) * -3]\).

Step 3 – Calculate the sum

Now, calculate the above expression to find the sum. \(S_{30} = 15\ [4 - 87] = 15 * -83 = -1245\)

Key Concepts

Sum of Arithmetic Sequences

An arithmetic sequence is a set of numbers in which each term after the first is generated by adding a constant value, known as the common difference, to the previous term. The sum of an arithmetic sequence involves adding all the terms together, which might seem challenging with bigger numbers or longer sequences. Here, a key concept comes in handy: There's a formula that makes this task more manageable. By using this formula, you can quickly find the total sum without needing to add up each individual term.

• The general approach is to determine the series' first and last terms.
• Use the formula which leverages these terms along with the number of terms.
• This avoids the cumbersome arithmetic of simple addition.

Understanding how to efficiently calculate the sum can greatly simplify many mathematical problems and applications.

First Term and Last Term in Sequences

Identifying the first term and the last term in an arithmetic sequence is crucial for calculating the sum correctly. In a sequence, the first term is essentially the starting point of your series. The last term is important as it marks the end of the series, especially when the sum of terms is being calculated.

• The formula given in the problem, \( -3i + 5 \), helps determine these values by substituting the smallest and largest numbers from your range into this formula.
• The first term, \(a_1\), often indicates the base value plus some number of complete steps based on the common difference.
• This continues until the last term, often the most negative or positive based on the common difference and sequence count.

Properly identifying these terms is the first step in solving many algebraic sequence problems.

Common Difference

The common difference in an arithmetic sequence is the constant value that separates each pair of consecutive terms. It's essentially what you add (or subtract, if negative) each time you move from one term to the next. Recognizing and calculating this difference is vital for fully understanding the pattern and structure of the sequence.

• You can find the common difference by subtracting the first term from the second.
• In our example, the formula for each term is \( -3i + 5 \), making \(-3\) the common difference.
• This consistent measure not only helps generate the terms but also feeds directly into the sum formula, impacting the calculation of the overall sequence.

Appreciating this difference helps in mapping how the sequence unfolds over its range of values.

Formula for Arithmetic Sequence Sum

The formula for finding the sum of an arithmetic sequence is an elegant tool that allows for quick and accurate calculations without the need to explicitly add every term. The formula is expressed as:\[S_n = \frac{n}{2} [2a + (n - 1)d]\]where:

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

This formula is grounded on the idea that the sum of the sequence can be calculated by considering both the count of terms and how they progress mathematically.
By appropriately substituting the right values, as shown in the problem, you can find the sequence sum efficiently, demonstrating the power of algebraic expressions in solving practical problems.