Problem 70 - Precalculus with Limits

Question

In Exercises 67 - 86, find the sum of the finite geometric sequence. \( \sum_{n=1}^{8}5\left(-\dfrac{5}{2}\right)^{n - 1} \)

Step-by-Step Solution

Answer

The sum of the finite geometric sequence is -39,320.

Step 1: Identification of Elements

Recognize and denote the first term as \( a = 5 \), the common ratio as \( r = -\dfrac{5}{2} \), and the number of terms as \( n = 8 \).

Step 2: Application of Formula

Apply the formula of the sum of a finite geometric series, which is \( S_n = a \dfrac{1 - r^n}{1 - r} \), and substitute the values of \( a \), \( r \), and \( n \) into the formula.

Step 3: Solving The Expression

Calculate \( S_8 = 5 \times \dfrac{1 - \left(-\dfrac{5}{2}\right)^8}{1 - \left(-\dfrac{5}{2}\right)} \), which simplifies to \( S_8 = -39,320 \).

Key Concepts

Sum of a Geometric Series

The sum of a geometric series is the total of all terms in a geometric sequence. A geometric sequence, to clarify, features a constant rate at which each term increases or decreases in relation to the one before it. This might seem complex, but it’s actually a straightforward concept once broken down. The key to finding the sum is by using a special formula designed for this type of series.

A real-world example could be anything that grows or shrinks by a certain percentage, like interest in a bank account, or even the spiraling seeds in a sunflower! Understanding how to calculate the sum can be really beneficial in these scenarios.

Through this concept, we can efficiently compute the collective value of terms without tediously adding each one, thanks to the formula specifically crafted for finite geometric series. In essence, it encapsulates the growth or decay of a sequence and summarizes it in a single, neat number.

Geometric Sequence

A geometric sequence is a string of numbers where each element after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. Imagine a pattern of blocks where each layer has twice as many blocks as the one below it. That increasing pattern of blocks is very much like a geometric sequence! If the blocks were getting smaller instead, that would represent a decreasing geometric sequence.

Mathematically, it's expressed as a set of numbers in this form: \( a, ar, ar^2, ar^3, ... \) where \( a \) is the first term and \( r \) is the common ratio. The common ratio is what gives the sequence its ‘geometric’ nature. Its value can result in a sequence that grows (if \( r > 1 \) or \( r < -1 \) ) or shrinks (if \( -1 < r < 1 \) ), which is clearly depicted in the growth or reduction rate of each term.

Common Ratio

The common ratio is the engine driving a geometric sequence forward. It’s defined as the factor by which consecutive terms multiply to generate the next one. To find it, you simply divide any term by its preceding element (it's a must that the sequence isn’t zero to start with, though).

This ratio could be negative or positive and can be a whole number, fraction, or an irrational number. When the ratio is more than 1, the sequence soars upward; less than 1 (but positive), it gently decreases. If the ratio flips to negative, expect the sequence to alternate with each step, swinging between positive and negative values. Knowing the common ratio is crucial, as it's central to determining how a sequence will behave and eventually, to finding the sum of its series.

Geometric Series Formula

Here's where math turns magic: the formula for the sum of a finite geometric series. It can be a lifesaver and time-saver! The formula looks like this: \[ S_n = a \frac{1 - r^n}{1 - r} \] where \( S_n \) is the sum of the first \( n \) terms, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

An interesting aspect of this formula is its behavior when dealing with common ratios between -1 and 1. In these cases, as \( n \) grows indefinitely, the \( r^n \) part of the formula gets closer and closer to zero, which means the formula for an infinite geometric series becomes even simpler. But for finite series, this formula enables you to calculate the sum efficiently, avoiding the manual addition of all terms. Just slot in the values, and voilà, you get the total sum needed!