Problem 53 - Algebra 2

Question

Which expression represents the sum of the finite series \(13+10+7+4 ?\) I. \(\sum_{n=1}^{4}(16-3 n)\) II. \(\sum_{n=3}^{6}(22-3 n)\) III. \(\sum_{n=1}^{4}(4+3 n)\) A. I and II B. I and III C. II and III D. I, II, and III

Step-by-Step Solution

Answer

Options I and II (i.e. answer choice A) are the correct representations of the given series sum. As computed, they both result in the sum 34, which is the sum of the given series. Option III gave a sum of 46, which does not match with the given series.

Step 1: Compute Sum for Option I

In this step, we will compute the sum of terms for Option I i.e., \(\sum_{n=1}^{4}(16-3 n)\). Let's replace n by 1, 2, 3, 4 and sum the results: (16-3*1) + (16-3*2) + (16-3*3) + (16-3*4) = 13 + 10 + 7 + 4 = 34, which equals the sum of the given series.

Step 2: Compute Sum for Option II

In this step, we will compute the sum of terms for Option II i.e., \(\sum_{n=3}^{6}(22-3 n)\). Let's replace n by 3, 4, 5, 6 and sum the results: (22-3*3) + (22-3*4) + (22-3*5) + (22-3*6) = 13 + 10 + 7 + 4 = 34, which also equals the sum of the given series.

Step 3: Compute Sum for Option III

In this step, we will compute the sum of terms for Option III i.e., \(\sum_{n=1}^{4}(4+3 n)\). Let's replace n by 1, 2, 3, 4 and sum the results: (4+3*1) + (4+3*2) + (4+3*3) + (4+3*4) = 7 + 10 + 13 + 16 = 46, which does not equal the sum of the given series.

Key Concepts

Understanding Summation Notation

Summation notation, often represented by the Greek letter sigma (\( \Sigma \)), provides a concise way to express the sum of a sequence of numbers. When using summation notation, we specify the range of summation and a general term of the sequence that is to be summed.

• The expression above the sigma indicates where the summation ends, while the expression below it defines where it starts.

For instance, the expression \( \sum_{n=1}^{4}(16-3n) \) means we start with \( n = 1 \) and go up to \( n = 4 \).
For each value of \( n \), we substitute \( n \) into the expression \( 16-3n \) and then add the results together.
This concept helps simplify the process of adding series, particularly when dealing with long sequences, by condensing the entire sum into a single, manageable expression.

Delving into Arithmetic Sequence

An arithmetic sequence is a sequence of numbers with a common difference between consecutive terms. In simple terms, you keep adding the same number to get from one term to the next.

• For example, the sequence \( 13, 10, 7, 4 \) decreases by 3 each step.

The general formula for any term in an arithmetic sequence is \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.
This formula is useful when you need to generate or identify terms in a sequence quickly.
Understanding and identifying arithmetic sequences is crucial since they form the backbone of many types of mathematical series.

Grasping Series Calculation

Series calculation involves finding the sum of terms in a sequence. In a finite series, like the one in the exercise, the number of terms is limited.
The goal is to sum the individual terms of a sequence.
Using the summation notation, arithmetic sequences can be easily summed up. For example, in our exercise:
  - For Option I, we calculated: \( 16-3 \times 1, 16-3 \times 2, 16-3 \times 3, 16-3 \times 4 \) and summed them to get \( 34 \).
  - Similarly, for Option II: \( 22-3 \times 3, 22-3 \times 4, 22-3 \times 5, 22-3 \times 6 \) also summed to \( 34 \).
Analyzing and comparing these results to the original sequence allows us to verify the correctness of our calculations.

• Option III did not match because its sum, created by the expression \( 4+3n \), resulted in a total of \( 46 \) instead of matching the original series sum.

Series calculation is fundamental for checking the equivalency between series expressions and understanding how different expressions represent the same mathematical object.