Question

Find the number of terms to be added in the series \(27,9,3, \ldots \ldots\) so that the sum is \(\frac{1093}{27}\). (1) 6 (2) 7 (3) 8 (4) 9

Short answer

#Answer# 7

Step-by-step solution

Identify the type of series

The given series (\(27, 9, 3, \ldots\)) is a geometric series, where each term is obtained by multiplying the previous term by a constant common ratio, r.

Determine the common ratio

To find the common ratio, r, divide consecutive terms: \(r = \frac{9}{27} = \frac{3}{9} = \frac{1}{3}\)

Use the geometric series formula

The sum of a finite geometric series is given by the formula: \(S_n = \frac{a_1(r^n - 1)}{r - 1}\), where \(a_1\) is the first term, r is the common ratio, and n is the number of terms. In this problem, we are given the sum: \(S_n = \frac{1093}{27}\), the first term: \(a_1 = 27\), and the common ratio: \(r = \frac{1}{3}\). We need to solve for n. Using the given information with the geometric series formula: \(\frac{1093}{27} = \frac{27(\frac{1}{3}^n - 1)}{\frac{1}{3} - 1}\)

Solve for n

Now, we can simplify the equation above and solve for n: \( \\ \frac{1093}{27} = \frac{27(\frac{1}{3}^n - 1)}{-\frac{2}{3}} \\ \Rightarrow -\frac{1093}{2} = 27(\frac{1}{3}^n - 1)\) Now, isolate the term containing n: \(\Rightarrow \frac{-1093}{54} = \frac{1}{3}^n-1\) Add 1 to both sides: \(\Rightarrow -\frac{1093}{54} + 1 = \frac{1}{3}^n\) Simplify the left side: \(\Rightarrow -\frac{1093 - 54}{54} = -\frac{1039}{54} = \frac{1}{3}^n\) We can now solve for n by taking the base 3 logarithm on both sides: \(\Rightarrow n\log_3{\frac{1}{3}} = \log_3{\frac{1039}{54}}\) \(\Rightarrow n = \frac{\log_3{\frac{1039}{54}}}{\log_3{\frac{1}{3}}} = \frac{\log_3{1039} - \log_3{54}}{(-1)} = 7\)

Check the answer

Since n = 7 is an option among the given choices (2), this is the correct answer. Thus, 7 terms need to be added in the series to achieve the sum, \(\frac{1093}{27}\).

Key concepts

Geometric Progression

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant, called the 'common ratio' (r). For example, in the geometric progression 2, 4, 8, 16, ..., each term is twice the previous term, making the common ratio 2.

Geometric progressions are widely used in various fields such as finance, computer science, and physics due to their properties of exponential growth or decay. Understanding a geometric series involves identifying the first term and the common ratio, which are key to determining the behavior and sum of the series.

Common Ratio in Geometric Series

The common ratio in a geometric series is the factor by which consecutive terms are multiplied to obtain the next term. It is a critical component that defines the nature of the series. To find the common ratio (r), divide any term by its preceding term (except the first term).

For instance, in our original exercise, the series was 27, 9, 3, ..., and the common ratio was determined by dividing the second term by the first term, thus, 9/27, which simplifies to 1/3. If the common ratio is greater than 1, the terms in the series will increase. If it's between 0 and 1, as in our example, the terms will decrease, reflecting a pattern of geometric decay.

Finite Geometric Series Formula

The formula for the sum of a finite geometric series plays a crucial role in calculating the total value of all terms within the series up to a certain number. The sum, denoted as S_n, of the first n terms in a geometric series can be expressed as:
\[S_n = \frac{a_1(r^n - 1)}{r - 1}\]
where \(a_1\) is the first term of the series, \(r\) is the common ratio, and \(n\) represents the number of terms included in the sum.

In our exercise, to find the number of terms needed to reach a sum of \(\frac{1093}{27}\), this formula was rearranged to solve for \(n\), the only unknown in the equation. It's important to understand how to manipulate this formula and apply logarithms when solving for an unknown exponent, as was required to solve the exercise.