Problem 8 - Algebra and Trigonometry

Question

For a geometric sequence with first term \(a_{1}\) and common ratio \(r,\) where \(r \neq 0, r \neq 1,\) the sum of the first \(n\) terms is \(S_{n}=a_{1} \cdot \frac{1-r^{n}}{1-r}\)

Step-by-Step Solution

Answer

Use \(S_n = a_1 \frac{1 - r^n}{1 - r}\) to find the sum of the first \(n\) terms of a geometric sequence.

Step 1: Identify the given terms

A geometric sequence is defined by its first term, denoted as \(a_1\), and its common ratio, denoted as \(r\). Given: \(r eq 0\) and \(r eq 1\).

Step 2: Understand the formula for the sum of the first n terms

The sum of the first \(n\) terms of a geometric sequence \(S_n\) is given by:\[ S_n = a_1 \frac{1 - r^n}{1 - r} \]This formula accounts for the series where \(r\) is not equal to 1 or 0.

Step 3: Break down the formula components

\(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. The numerator \(1 - r^n\) captures the effect of the ratio over \(n\) terms, and the denominator \(1 - r\) normalizes the sum.

Step 4: Substitute values (if given)

If specific values for \(a_1\), \(r\), and \(n\) are provided, substitute them into the formula to find the sum \(S_n\).

Step 5: Simplify the expression

After substitution, simplify the expression to find the sum.

Key Concepts

sum of series

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. We often need to find the sum of a given number of terms in such a sequence. The sum of the first n terms of a geometric series is given by the formula:
\[ S_n = a_1 \frac{1 - r^n}{1 - r} \]
Here, \(S_n\) is the sum of the first n terms, \(a_1\) is the first term, and \(r\) is the common ratio. This formula helps us quickly compute the sum without having to add each term individually.

common ratio

The common ratio, denoted as \(r\), is a crucial part of a geometric sequence. It is the factor by which we multiply each term to get the next term in the sequence. For example, in the sequence 2, 6, 18, 54, ... the common ratio is 3 because each term is 3 times the previous one.
The common ratio must not be 0 or 1:

• If \(r = 0\), the sequence would just be zeros after the first term.
• If \(r = 1\), the sequence would just repeat the first term.

Understanding the common ratio helps in using the formula for the sum of the terms effectively.

first term

The first term of a geometric sequence, typically denoted as \(a_1\), plays a significant role in determining the sequence's behavior. It is the starting point of the sequence. For example, in the sequence 5, 10, 20, 40, ... the first term \(a_1\) is 5.
Knowing the first term is essential because it is used directly in the sum formula. The sum of the first n terms depends on this term, as it is multiplied by a factor to account for the common ratio and the number of terms.

formula derivation

Deriving the formula for the sum of the first n terms of a geometric sequence involves a few steps. Here's how you can understand it:

• Start with the geometric sequence: \(a_1, a_1r, a_1r^2, ..., a_1r^{n-1}\).
• Write the sum of these terms: \(S_n = a_1 + a_1r + a_1r^2 + ... + a_1r^{n-1}\).
• To derive the formula, multiply both sides of the equation by the common ratio \(r\):
  \(rS_n = a_1r + a_1r^2 + a_1r^3 + ... + a_1r^n\).
• Subtract the second equation from the first:
  \(S_n - rS_n = a_1 - a_1r^n\).
• Factor out \(S_n\) and \(a_1\):
  \(S_n(1 - r) = a_1(1 - r^n)\).
• Solve for \(S_n\):
  \[ S_n = a_1 \frac{1 - r^n}{1 - r} \]

And that's the derived formula to calculate the sum of the first n terms in a geometric sequence!