Problem 32 - Algebra and Trigonometry

Question

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence. $$ \sum_{i=1}^{6} 4^{i} $$

Step-by-Step Solution

Answer

The sum of the first 6 terms of the geometric sequence is 5460.

Step 1: Identify the first term (a), the common ratio (r), and the number of terms (n)

From the sigma notation, the first term \(a = 4^1 = 4\), the common ratio \(r = 4\), and the number of terms \(n = 6\).

Step 2: Substitute a, r and n into the formula for the sum of the first n terms of a geometric sequence

Substitute \(a=4\), \(r=4\), and \(n=6\) into the formula \(S_n = \frac{a(r^n - 1)}{r-1}\). This gives \(S_6 = \frac{4(4^6 - 1)}{4-1}\).

Step 3: Calculate the sum

Simplify the expression. \(S_6 = \frac{4(4096 - 1)}{3} = \frac{4 × 4095}{3} = 5460.\)

Key Concepts

Sum of Geometric Sequence

The sum of a geometric sequence is the total of all the terms in the sequence. When you have a geometric sequence and want to find this sum, there's a handy formula you can use: \[ S_n = \frac{a(r^n - 1)}{r-1} \] Here,

• \(S_n\) is the sum of the first \(n\) terms.
• \(a\) is the first term.
• \(r\) is the common ratio.
• \(n\) is the number of terms.

This formula simplifies finding the sum without having to manually add each term. By plugging the values into this formula, you can quickly calculate the sum of a geometric sequence. This technique is particularly useful when the number of terms is large.

Common Ratio

The common ratio is a crucial element in a geometric sequence. It tells us how each term is related to the one before it by expressing the factor by which we multiply one term to get the next. For the sequence described in the original exercise, the common ratio \(r\) is 4. If you want to determine the common ratio in any geometric sequence, you just need to take any term (except the first one) and divide it by the term immediately preceding it.
This ratio remains constant throughout the sequence, which is exactly why these sequences are called 'geometric'. It's this consistent scaling by the common ratio that creates the unique properties of geometric sequences.

First Term

The first term of a geometric sequence is where it all begins. It’s the starting point from which all other terms are derived. In our original example, the first term \(a\) was determined to be 4, as the sequence starts with \(4^1\). Knowing this first term is essential because it serves as the base value that is repeatedly multiplied by the common ratio to generate each subsequent term. Whenever you're going to use the formula for the sum of a geometric sequence, identifying the first term is your first step.

Number of Terms

The number of terms, denoted by \(n\), specifies how many terms in the sequence are being considered. In the provided exercise, \(n\) is equal to 6, meaning that we are summing up the first six terms of the sequence. Identifying how many terms to include in your calculations is important.
It directly affects the sum of the sequence because it establishes the limit of the sequence's progression, essentially determining where the summation stops. The number of terms \(n\) will affect the power to which you raise the common ratio when using the sum formula, making it a key concept for accurate calculations.