Question

Find the sum of \(\frac{0.3}{0.5}+\frac{0.33}{0.55}+\frac{0.333}{0.555}+\ldots \ldots \ldots \ldots\) to 15 terms. (1) 10 (2) 9 (3) 3 (4) 5

Short answer

Answer: The sum of the given series up to 15 terms is 9.

Step-by-step solution

Identify the pattern

We can see that both the numerators and denominators of the terms in the series have a common pattern: The numerators are formed by adding more 3's while the denominators are formed by adding more 5's. We can rewrite the terms as: \(\frac{3\cdot10^{-1}}{5\cdot10^{-1}}\) , \(\frac{33\cdot10^{-2}}{55\cdot10^{-2}}\) , \(\frac{333\cdot10^{-3}}{555\cdot10^{-3}}\) , \(\dots\)

Find the general term

Using the discovered pattern, we can write the generic term \(T_n\) of the series as follows: \(T_n = \frac{[(10^n-1)\cdot3]\cdot10^{-n}}{[(10^n-1)\cdot5]\cdot10^{-n}}\) Simplifying, it becomes: \(T_n = \frac{3(10^n-1)}{5(10^n-1)}\) Both the numerator and the denominator share a common factor \((10^n-1)\). We can divide the numerator and the denominator by this factor: \(T_n = \frac{3}{5}\)

Find the sum of the first 15 terms

Since we found that the generic term \(T_n=\frac{3}{5}\), this means that all the terms in the series are equal to \(\frac{3}{5}\). As a result, we can find the sum of the first 15 terms by simply multiplying \(\frac{3}{5}\) by 15: \(S_{15} = 15 \cdot \frac{3}{5}\)

Calculate the final answer

Multiply the fraction by 15: \(S_{15} = 15 \cdot \frac{3}{5} = \frac{45}{5}\) And after simplification, we have: \(S_{15} = 9\) The sum of the given series up to 15 terms is 9, which corresponds to option (2).

Key concepts

Pattern Recognition

Recognizing patterns is a key skill in mathematics. For the given series, the pattern emerges in the structure of the numerators and denominators. Each successive term in the sequence builds upon the last by adding additional 3s and 5s. This kind of pattern is referred to as a sequence defined by repetition of numerals.
To identify this pattern:

• Notice that the numerators progress through numbers such as 0.3, 0.33, 0.333, and so forth.
• Similarly, the denominators follow numbers like 0.5, 0.55, 0.555, etc.

Recognizing these repetitive elements helps simplify the formulation and calculations of the series. Once this pattern is understood, it is easier to manipulate these terms in a structured mathematical form.

General Term Calculation

General term calculation involves establishing a formula that can generate any term in the series. Using the observed pattern in the sequence, we set up the general term, denoted by \(T_n\). The aim is to express each term consistently with respect to its sequence number, \(n\).
Based on the pattern, we derive:

• Numerators: \((10^n - 1) \cdot 3 \cdot 10^{-n}\)
• Denominators: \((10^n - 1) \cdot 5 \cdot 10^{-n}\)

This representation captures the essence of our pattern in a simplified mathematical formula. By simplifying further, these terms reduce to \(\frac{3}{5}\). Establishing this formula is crucial, as it sets the stage for further calculations like the summation of the series.

Fraction Simplification

Simplifying fractions is essential to make expressions easier to work with. In this case, both the numerator and the denominator of our general term \(T_n\) have a common factor, \((10^n-1)\). By dividing both by this factor, the term simplifies to \(\frac{3}{5}\).
Fraction simplification involves:

• Identifying a common factor in both the numerator and denominator.
• Dividing both by this common factor to reduce the fraction to its simplest form.

Simplification makes it easier to handle calculations without carrying undue complexity through subsequent steps. Here, all terms in the series simplify consistently to \(\frac{3}{5}\), highlighting the uniform nature of this arithmetic series.

Term Summation

The summation of sequence terms is straightforward once each is equal. With each term simplified to \(\frac{3}{5}\), we can easily find the sum of the first 15 terms by multiplying: The number of terms by the simplified value of each term.
The formula is simple:

• Sum \(S_{15} = 15 \cdot \frac{3}{5}\)

Perform the multiplication:

• \(S_{15} = \frac{45}{5} = 9\)

This final result demonstrates that adding consistent fractions in an arithmetic series is efficient and results in a straightforward calculation. Mastering term summation allows you to find the result without needing to add each term separately.