Problem 40 - Algebra and Trigonometry

Question

Find the sum of each infinite geometric series. $$ 5+\frac{5}{6}+\frac{5}{6^{2}}+\frac{5}{6^{3}}+\cdots $$

Step-by-Step Solution

Answer

The sum of this infinite geometric series is 6.

Step 1: Identify the first term

The first term \(a\) of the series is the first number of the series, which is 5.

Step 2: Identify the common ratio

The common ratio \(r\) is found by dividing any term in the sequence by the previous term. Here, it is \(\frac{5}{6}\) divided by 5 which is \(\frac{1}{6}\).

Step 3: Apply the formula

Now we can apply the formula for the sum of an infinite geometric series: \(S=\frac{a}{1-r}\). Substituting the values a=5, r=\(\frac{1}{6}\) into the formula, we get \(S=\frac{5}{1-\frac{1}{6}}\)

Step 4: Simplify the expression

Simplifying the expression gives \(S=\frac{5}{\frac{5}{6}}\). Thus, S=6

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term is obtained by multiplying the previous one by a constant, known as the common ratio. In our exercise, the sequence begins with 5 and continues indefinitely. Each term follows this pattern:

• The second term is \( \frac{5}{6} \).
• The third term is \( \frac{5}{6^2} \).
• The fourth term is \( \frac{5}{6^3} \).

This regular pattern continues forever.
The geometric nature means a consistent multiplication process at work, distinguishing it from an arithmetic sequence where addition is the rule. Instead, in a geometric sequence, the multiplicative nature drives the progression of terms.

Sum of Series

The sum of an infinite geometric series depends on the series' properties, specifically having a common ratio smaller than 1 in absolute value. This ensures the terms diminish, allowing the total sum to converge to a finite limit.
In the given series, using the sum formula, the convergence means no matter how many terms you add, the series approaches a precise number, not infinity. The sum of our infinite series using the formula \( S = \frac{a}{1-r} \) yields a clean result, ensuring convergence and allowing practical calculation.

Common Ratio

The common ratio is the backbone of any geometric sequence, determining how each term relates to the one before it. Calculated by dividing any term by its predecessor, it remains consistent across the sequence.
In the example, dividing \( \frac{5}{6} \) by 5 results in \( \frac{1}{6} \). This consistent pattern facilitates the sequential multiplication that defines geometric sequences. Because this ratio is less than 1, it ensures that each subsequent term in our series decreases in magnitude, a crucial factor allowing the series to converge to a sum.

First Term

The first term in a geometric series is the starting point from which all other terms are derived. It sets the series in motion and is represented symbolically by \( a \).
In our problem, the first term is straightforwardly identified as 5, the initial term given.
This value is essential in applying formulas like the sum of the series formula, \( S = \frac{a}{1-r} \), which requires knowing the starting point to calculate the total sum accurately. It serves as the base of the sequence from which every subsequent term is generated, relying heavily on the common ratio.