Calculate The Sum of The Series 6-4 6 N

What is the sum of a series?

A series is the sum of the terms of a sequence. For example, the series 6-4 6 n represents a sequence where each term is calculated based on the previous term. The sum of a series is the total when you add all its terms together.

There are different types of series, including arithmetic, geometric, and others. The method for calculating the sum depends on the type of series. For the series 6-4 6 n, we'll use the arithmetic series formula.

Formula for the series sum

The sum S of the first n terms of an arithmetic series can be calculated using the formula:

Where:

• S is the sum of the series
• n is the number of terms
• a₁ is the first term
• d is the common difference between terms

For the series 6-4 6 n, we can identify:

• First term (a₁) = 6
• Common difference (d) = -4
• Number of terms (n) = variable

Worked example

Let's calculate the sum of the first 5 terms of the series 6-4 6 n.

1. Identify the values:
   • a₁ = 6
   • d = -4
   • n = 5
2. Plug the values into the formula:
   S = 5/2 × (2×6 + (5-1)×-4)
3. Calculate inside the parentheses:
   2×6 = 12 (5-1)×-4 = 4×-4 = -16 12 + (-16) = -4
4. Multiply by n/2:
   5/2 × -4 = 2.5 × -4 = -10

The sum of the first 5 terms is -10.

Practical applications

Calculating the sum of a series is useful in various fields:

• Finance: Calculating the present value of an annuity
• Physics: Determining the total displacement from a series of movements
• Computer Science: Summing elements in an array
• Engineering: Calculating total force or energy over time

Understanding how to calculate series sums helps in solving real-world problems involving sequences and patterns.