Problem 35 - Algebra and Trigonometry

Question

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$ \sum_{i=1}^{6}\left(\frac{1}{2}\right)^{i+1} $$

Step-by-Step Solution

Answer

The sum of the first 6 terms of the sequence is \(\frac{63}{128}\).

Step 1: Identify the First Term, Common Ratio, and Number of Terms

Looking at the problem, we can see that the first term \(a\) is obtained when \(i = 1\), which gives us \((\frac{1}{2})^{1+1} = \frac{1}{4}\). The common ratio \(r\) is the base of the exponent, which is \(\frac{1}{2}\). The number of terms \(n\) is given as 6.

Step 2: Plug the Values into the Formula

Now that we have \(a = \frac{1}{4}\), \(r = \frac{1}{2}\), and \(n = 6\), we plug these values into the formula: \(S_n = \frac{1}{4} \cdot \frac{1 - (\frac{1}{2})^6}{1 - \frac{1}{2}}.

Step 3: Simplify the Equation

Solving the equation gives us the value of the sum: \(S_6 = \frac{1}{4} \cdot \frac{1 - \frac{1}{64}}{\frac{1}{2}} = \frac{1}{2} \cdot \frac{1 - \frac{1}{64}} = \frac{63}{128}\).

Key Concepts

Geometric Progression

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For instance, in the sequence 2, 4, 8, 16, ..., the common ratio is 2 because each term is twice the preceding term.

The formula to find the nth term of a geometric progression is given as:\[ a_n = a_1 \times r^{(n-1)} \]where \( a_n \) is the nth term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. This formula helps you determine any term in the sequence without having to list all the previous terms.

Sum of n Terms

The sum of the first \( n \) terms of a geometric progression can be calculated using a specific formula. The sum is represented as \( S_n \) and is given by:\[ S_n = a_1 \times \frac{1 - r^n}{1 - r} \]if \( r \) is not equal to 1. Here, \( a_1 \) is the first term and \( r \) is the common ratio. The numerator deals with the growth factor, subtracting the compound effect of the ratio after \( n \) terms from 1. The denominator subtracts the common ratio from 1 to normalize this effect for a summation. It's critical to handle the formula correctly; an incorrect ratio or number of terms can lead to the wrong sum.

Geometric Series

A geometric series is the sum of the terms of a geometric progression. For example, adding the terms of the sequence 2, 4, 8, 16, ..., we get a geometric series. The series can be finite or infinite, depending on how many terms are added up. In the case of an infinite series, the sum approaches a finite limit if the absolute value of the ratio is less than 1; this is known as a convergent series.

An example of a sum for a finite geometric series is the exercise initially given, where you calculate \( \sum_{i=1}^{6}\left(\frac{1}{2}\right)^{i+1} \), which represents the sum of the first 6 terms of a geometric progression with a common ratio of \( \frac{1}{2} \).

Sequence and Series

The concepts of sequence and series are foundational in understanding different types of numerical patterns. A sequence is an ordered list of numbers, while a series is the sum of a sequence of numbers. Notably, sequences can be either finite or infinite, and they're defined by a rule that predicts the next number given the current terms.

When dealing with series, it's also important to distinguish between arithmetic and geometric series. An arithmetic series has a constant difference between consecutive terms, while a geometric series has a constant ratio. The problem-solving approach varies significantly between these two types, as they follow different patterns and formulae for calculating sums or individual terms.