Problem 103 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Show that the numbers \(10,11,12\) cannot be the terms of a single G.P. with common ratio not equal to \(1 .\)

Step-by-Step Solution

Answer

The numbers \(10, 11, 12\) can't form a geometric progression with common ratio not equal to 1, because the ratios \(11/10\) and \(12/11\) are not equal.

Step 1: Calculate first ratio

First, we calculate the ratio between the second term and the first term. Hence, the first ratio, \(r1 = 11 / 10\)

Step 2: Calculate second ratio

Next, we calculate the ratio between the third term and the second term. Hence, the second ratio, \(r2 = 12 / 11\)

Step 3: Comparison of Ratios

Finally, we compare the two ratios, \(r1\) and \(r2\). If they are equal, this would mean that the numbers form a G.P., however the problem states that the common ratio is not equal to 1. Checking the values of \(r1\) and \(r2\), we see that \(r1 = 1.1\) while \(r2 = 1.09\). Since \(r1\) and \(r2\) are not equal, we can say that \(10,11,12\) do not form a G.P.

Key Concepts

Common Ratio

Understanding the common ratio is crucial when dealing with geometric progressions (G.P.). A geometric progression 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. The common ratio has remarkable consistent qualities across a G.P. For instance, in a sequence like \( 2, 6, 18, ... \), the common ratio is \( 6/2 = 3 \).

This consistent multiplication is what sets a G.P. apart from other sequences. It’s worth noting that the common ratio can be any real number, positive or negative, except zero, as a ratio of zero would make all succeeding terms in the sequence zero, which is not practical. Extra attention should be paid when working with negative ratios, as they cause the sequence to alternate between positive and negative numbers, an essential characteristic to recognize in sequence comparison.

Sequence Comparison

Sequence comparison plays a pivotal role in identifying the type of sequence we are dealing with. Comparing geometric sequences involves looking at the ratios of successive terms. If these ratios are consistent—that is, they remain the same throughout the sequence—then the sequence can be identified as a geometric progression.

However, if the ratios vary, then the sequence is not a geometric progression. For example, if we have a sequence \( a, b, c \), and we calculate \( b/a \) and \( c/b \) and these ratios are not equal, then we can conclusively say that \( a, b, c \) do not form a G.P. This kind of comparison is instrumental when analyzing sequences, as it provides a clear, mathematical way to validate or refute whether a specific sequence follows a geometric pattern.

Ratio Calculation

Ratio calculation is an essential step when working with geometric progressions, as it helps to determine whether a sequence of numbers can constitute a G.P. Given a sequence, to calculate the ratio, we simply take any term after the first term and divide it by the term immediately preceding it. The process is repeated with multiple terms to ensure the ratio is constant.

In practical terms, with a sequence such as \( 5, 10, 20 \), the ratio is calculated by doing \( {10}/{5} \) and then \( {20}/{10} \), both of which yield \(2\), indicating the sequence is indeed a G.P. with a common ratio of \(2\). Applying this method systematically is central for correctly identifying geometric progressions and understanding their underlying structures.