Problem 47 - College Algebra

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 the first 30 terms of the sequence is -1245.

Step 1: Finding the First Three Terms

To find the first three terms, we substitute the first three integers into the sequence equation. So, for i = 1, 2, 3, the first three terms respectively would be: \(-3*1 + 5 = 2\), \(-3*2 + 5 = -1\), \(-3*3 + 5 = -4\).

Step 2: Finding the Last Term

To find the last term, we substitute the last integer in the sequence, which is 30. So, for i = 30: \(-3*30 + 5 = -85\).

Step 3: Finding the Sum of First N Terms

We know in an arithmetic sequence, the formula to find the sum of the first n terms (S) is given as: \(S = n/2 * (a + l)\), where 'n' is the number of terms, 'a' is the first term and 'l' is the last term. Substituting the respective numbers into the equation: \(S = 30/2 * (2 - 85) = -1245\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. This difference is called the common difference and is denoted as 'd'. For instance, in the sequence 2, 4, 6, 8, ..., the common difference 'd' is 2 since each term is obtained by adding 2 to the prior term.

To illustrate this with the exercise provided, we first identify the common difference by looking at the change from one term to the next. In this case, each term decreases by 3, so our common difference 'd' is -3. This common difference is crucial for understanding the pattern of the sequence and is used when we want to find the sum of a certain number of terms.

Sequence Formula

For an arithmetic sequence, the nth term can be found using the sequence formula: \( a_n = a_1 + (n - 1)d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, and 'd' is the common difference. This formula quickly allows us to find any term in the sequence.

In the given exercise, the sequence formula \( -3i + 5 \) determines the value of the ith term. If 'i' is replaced with a specific number, like 1, 2, or 30 (as done in the exercise), it provides the value of the 1st, 2nd, or 30th term, respectively. By plugging in different values of 'i', we can find the corresponding terms without having to list the entire sequence.

Sum of Series

Calculating the sum of an arithmetic sequence involves adding all the terms up to the nth term. The sum of the first n terms of an arithmetic sequence can be found using the formula: \( S = \frac{n}{2} (a_1 + a_n) \), where \( S \) is the sum, \( n \) is the number of terms, \( a_1 \) is the first term and \( a_n \) is the nth term. Another variant of the formula uses the common difference 'd' and is written as \( S = \frac{n}{2} (2a_1 + (n - 1)d) \).

For the exercise, using the provided sum formula, we efficiently calculate the sum of the first 30 terms without having to manually add each term, which can be time-consuming. The formula is a fundamental tool in arithmetic series and showcases the beauty of mathematical shortcuts.

Mathematical Induction

Mathematical induction is a technique for proving that a statement, hypothesis, or formula is true for all natural numbers. It consists of two steps: the base case and the inductive step. In the base case, you prove that the statement is true for the first natural number, often 1. In the inductive step, you assume the statement is true for an arbitrary natural number 'k' and then prove it is true for 'k+1'.

While not explicitly used in the example exercise, mathematical induction can be a valuable tool when dealing with proofs in sequences and sums. For arithmetic sequences, one might use induction to prove properties like the sequence formula or the sum formula.