91Ó°ÊÓ

A finite geometric series is a series with a finite number of terms, where each term after the first is found by multiplying the previous term by a constant called the 'common ratio'. For example, in the series given in the exercise:

• First term, \(a = 1\)
• Second term is \(ar = 1 \times \frac{1}{2} = \frac{1}{2}\)
• Third term is \(ar^2 = 1 \times (\frac{1}{2})^2 = \frac{1}{4}\)
• Fourth term is \(ar^3 = 1 \times (\frac{1}{2})^3 = \frac{1}{8}\)

This pattern continues for a finite number of terms.

To find the sum of the first \(n\) terms of a finite geometric series, we use the sum formula: equation: \( S_n = a \frac{1 - r^n}{1 - r} \). In this formula:

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

For example, to find the sum of the given series with the first term \(1\), common ratio \(\frac{1}{2}\), and four terms, we plug these values into the formula.

Simplifying mathematical expressions is a crucial step in finding the solution. For the sum formula, we need to calculate and substitute the values correctly. First, we substitute \(a = 1\), \(r = \frac{1}{2}\), and \(n = 4\) into the sum formula: equation: \( S_4 = 1 \frac{1 - (\frac{1}{2})^4}{1 - \frac{1}{2}} \). Next, we calculate \((\frac{1}{2})^4\): equation: \((\frac{1}{2})^4 = \frac{1}{16}\) Then, substitute back into the formula: equation: \(S_4 = \frac{1 - \frac{1}{16}}{1 - \frac{1}{2}}\) Simplify the numerator: equation: \(1 - \frac{1}{16} = \frac{16}{16} - \frac{1}{16} = \frac{15}{16}\) Finally, divide to get the sum: equation: \(S_4 = \frac{\frac{15}{16}}{\frac{1}{2}} = \frac{15}{16} \times 2 = \frac{30}{16} = \frac{15}{8}\).

The common ratio in a geometric series is the factor by which each term after the first is multiplied to get the next term. It is a crucial element in determining the series and applying the sum formula.In the series \(1, \frac{1}{2}, (\frac{1}{2})^2, (\frac{1}{2})^3\), the common ratio \(r\) is \(\frac{1}{2}\). To identify the common ratio:

• Divide the second term by the first term: \(\frac{\frac{1}{2}}{1} = \frac{1}{2}\)
• Divide the third term by the second term: \(\frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2}\)
• Check that this pattern holds for subsequent terms

Understanding the common ratio helps in applying the sum formula correctly and solving geometric series problems effectively.