Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the sum. $$\sum_{i=1}^{6}(3 i-1)$$

Step-by-Step Solution

Answer

The sum of the series is 57.

Step 1: Calculate a_1 and a_n

To begin with, calculate the first term (a_1 = 3*1 - 1 = 2) and the last term (a_n = a_6 = 3*6 - 1 = 17).

Step 2: Insert a_1 and a_n into the Sum formula

We will now use the sum of arithmetic series formula which is \(S_n = \frac{n}{2}(a_1 + a_n)\). Here, \(n = 6\), \(a_1 = 2\) and \(a_n = a_6 = 17\). Insert these into the formula to calculate the sum.

Step 3: Compute the sum

By substituting the values, you obtain \(S_n = \frac{6}{2}(2 + 17)\), which simplifies to \(S_n = 3*(19)\ = 57\). This is the sum of the series.

Key Concepts

Sum Formula

In arithmetic series, a common way to find the sum is to use the sum formula. This formula is handy for quickly calculating the total of a sequence without adding each term individually. The sum formula for an arithmetic series is \( S_n = \frac{n}{2}(a_1 + a_n) \), where:

• S_n is the sum of the first n terms in the series
• n is the number of terms
• a₁ is the first term
• a_n is the last term

Using this formula, you can easily find the sum by knowing the first and last terms and how many terms there are. For example, to find the sum of the series \( \sum_{i=1}^{6}(3i-1) \), we calculate the first term \(a_1 = 2\) and the last term \(a_6 = 17\). With 6 terms in the sequence, you substitute into the formula to get \( S_6 = \frac{6}{2}(2 + 17) = 57 \). This shows how efficiently the sum formula works.

Sequence Terms

In any sequence, understanding the terms is crucial. A sequence is composed of individual elements, often generated by a specific rule or formula. Each element in a sequence is referred to as a "term."
For the arithmetic sequence \( \sum_{i=1}^{6}(3i-1) \), each term is calculated by substituting values of i from 1 to 6 into the expression \(3i-1\).
This gives:

• a₁ = 2 (when i=1)
• a₂ = 5 (when i=2)
• a₃ = 8 (when i=3)
• a₄ = 11 (when i=4)
• a₅ = 14 (when i=5)
• a₆ = 17 (when i=6)

Identifying these individual terms helps in understanding the progression and calculating the sum for any arithmetic sequence.

Arithmetic Progression

Arithmetic progression means that each term after the first is found by adding a constant difference to the previous term. This constant difference is called the "common difference."
For the sequence \( \sum_{i=1}^{6}(3i-1) \), you find that each term increases by 3 (i.e., 5-2 = 3, 8-5 = 3, and so on).
This consistent step is what defines an arithmetic progression.

• First term (\(a_1\)): 2
• Common difference (\(d\)): 3
• Next term (\(a_2\)): 5

Each new term is obtained by adding this common difference to the previous term. Recognizing this pattern helps in identifying series and using efficient ways to sum them, such as the sum formula. Understanding the concept of arithmetic progressions is essential in solving many mathematical problems relating to sequences.