Problem 276 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Sum up to \(n\) terms the series \(0.7+0.77+0.777+\ldots\)

Step-by-Step Solution

Answer

The sum of the first \(n\) terms of the series is given by \(S_n = 0.7 \cdot \frac{1-(1.1)^n}{1-1.1}\).

Step 1: Identify the first term and the common ratio

The first term \(a\) of the series is 0.7 and the common ratio \(r\) can be found by dividing the second term by the first term. In this case, \(r = 0.77/0.7 = 11/10 = 1.1\).

Step 2: Find the nth term

The nth term \(T_n\) of a geometric series can be found as follows: \(T_n = a \cdot r^{n-1}\). Computing the general form for this series, we get: \(T_n = 0.7 \cdot (1.1)^{n-1}\).

Step 3: Calculate the sum

The sum \(S_n\) of the first n terms of a geometric series is given by the formula: \(S_n = a \cdot \frac{1-r^n}{1-r}\). Hence, \(S_n = 0.7 \cdot \frac{1-(1.1)^n}{1-1.1}.\)

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 one by a fixed, non-zero number called the common ratio. For instance, in the series given in the exercise, which starts with 0.7, 0.77, 0.777, and so on, each term is generated by multiplying the previous term by 1.1.

Identifying the common ratio is crucial as it defines the pattern of the series. For a series to qualify as a geometric progression, this ratio must remain consistent throughout the series. The common ratio affects the growth rate of the series; if it is greater than 1, the series grows, and if it's between 0 and 1, the series decreases. Negative ratios cause the series to oscillate between positive and negative values. Understanding the common ratio allows for the calculation of any term within the sequence as well as the sum of terms, which are fundamental aspects of understanding geometric progressions.

Nth Term of Geometric Series

The nth term of a geometric series is the term that occupies the nth position in the sequence. It can be calculated using the formula:
\( T_n = a \cdot r^{n-1} \) where \( T_n \) is the nth term, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

In our example, with a first term of 0.7 and a common ratio of 1.1, you would find the nth term of the series by placing these values into the formula above. This concept allows for the direct calculation of any term within a geometric progression without having to list all preceding terms, which becomes particularly useful for series with a large number of terms. Navigating through the geometric series using this nth term formula enables students to deeply understand the progression's behavior and to solve related problems efficiently.

Sum of Geometric Series

The sum of a geometric series up to a particular number of terms, denoted as \( S_n \), is found using the formula:
\( S_n = a \cdot \frac{1-r^n}{1-r} \) where \( S_n \) is the sum of the first n terms, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms to be summed. This formula is effective only when \( r eq 1 \), as the denominator would be zero, which is not mathematically valid.

Applying this formula to the exercise provided, the sum of the first n terms of the series starting with 0.7 and with a common ratio of 1.1 is calculated. It is a crucial tool for students working with geometric series in various applications such as finance, computer science, and physics. By understanding how to apply this formula correctly, students can efficiently determine the sum of a geometric series without laboriously adding each term.