Problem 85 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If, in a G.P. of \(3 n\) terms, \(S_{1}\) denotes the sum of the first \(n\) terms, \(S_{2}\) the sum of the second block of \(n\) terms and \(S_{3}\) the sum of the last \(n\) terms, then \(S_{1}, S_{2}, S_{3}\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

Step-by-Step Solution

Answer

(B) S1, S2, S3 are in a G.P.

Step 1: Understand the Problem

We need to determine the relationship between the sums of three segments \(S_1, S_2, S_3\) in a geometric progression consisting of \(3n\) terms. Specifically, we will analyze the sum of the first \(n\) terms, the second block of \(n\) terms, and the last \(n\) terms.

Step 2: Define the Terms of the G.P.

Let the first term of the G.P. be \(a\) and the common ratio be \(r\). The terms of the G.P. are \(a, ar, ar^2, \dots, ar^{3n-1}\).

Step 3: Calculate \(S_1\)

\(S_1\) is the sum of the first \(n\) terms of the G.P.:\[S_1 = a + ar + ar^2 + \ldots + ar^{n-1} = a \frac{r^n - 1}{r - 1}\]

Step 4: Calculate \(S_2\)

\(S_2\) is the sum of the next \(n\) terms (terms \(n+1\) to \(2n\)):\[S_2 = ar^n + ar^{n+1} + ar^{n+2} + \ldots + ar^{2n-1} = ar^n \frac{r^n - 1}{r - 1}\]

Step 5: Calculate \(S_3\)

\(S_3\) is the sum of the last \(n\) terms (terms \(2n+1\) to \(3n\)):\[S_3 = ar^{2n} + ar^{2n+1} + \ldots + ar^{3n-1} = ar^{2n} \frac{r^n - 1}{r - 1}\]

Step 6: Analyze the Relationship Between \(S_1, S_2, S_3\)

From the previous steps:\[ S_2 = r^n S_1 \quad \text{and} \quad S_3 = r^{2n} S_1 \]From these equations, the sequence \(S_1, S_2, S_3\) forms a geometric progression with a common ratio \(r^n\).

Step 7: Conclude the Pattern

Since \(S_2 = r^n S_1\) and \(S_3 = r^{2n} S_1\), \(S_1, S_2, S_3\) are in a geometric progression (G.P.).Thus, the answer is (B) G.P.

Key Concepts

Arithmetic Progression

Arithmetic Progression (A.P.) refers to a sequence of numbers in which the difference between any two consecutive terms is constant. Imagine a stairway where each step consistently rises by the same amount each time you take a step. This consistent rise or difference is called the "common difference."

• Example: 2, 4, 6, 8, ... with a common difference of 2.
• Formula for the nth term: If the first term is denoted as \(a\) and the common difference is \(d\), then the nth term \(a_n\) is \(a + (n-1)d\).

Understanding arithmetic progression is crucial when comparing it with other types of progressions, such as geometric progression. While in A.P., we add or subtract the same value repeatedly, in a G.P., we multiply or divide by a constant factor.

Sum of Series

The sum of the terms in a sequence can be referred to as a sum of series. Depending on the type of sequence (arithmetic or geometric), different formulas are used. In the case of a geometric progression (G.P.), which is the focus here, the series sums up in powers.

• Geometric Sum Formula: For a G.P. with first term \(a\), common ratio \(r\), and \(n\) terms, the sum \(S_n\) is \( a \frac{r^n - 1}{r - 1} \) if \(r eq 1\).
• Arithmetic Sum Formula: For an A.P. with first term \(a\), last term \(l\), and \(n\) terms, the sum \(S_n\) is \( \frac{n}{2}(a + l) \).

The knowledge of finding a series sum helps in analyzing whole segments of progressions, providing insights into their behavior over multiple terms. In the given exercise, knowing how to sum each segment of terms in a G.P. was vital to determine their overall relationship.

Common Ratio

A common ratio in a geometric progression is a fundamental aspect that determines the progression's behavior. It is the consistent factor by which we multiply each term to get to the next. Recognizing this factor can quickly help identify the type and properties of the sequence.

• For instance, in the sequence 3, 6, 12, 24, ..., the common ratio \(r\) is 2.
• Formula: If \(a\) is the first term and it grows to the next term \(ar\), then \(r = \frac{a_2}{a_1}\).

Knowledge of the common ratio provides essential information for exploring if a sequence is in G.P., as was needed in the exercise. It also supports distinguishing it from arithmetic progression, where no multiplication by a constant factor is involved, just addition or subtraction.

Harmonic Progression

Harmonic Progression (H.P.) is less common than arithmetic or geometric progressions but is still vital in certain contexts. This progression is formed by taking the reciprocals of an arithmetic progression.

• For example, consider the A.P. 2, 4, 6, 8; the H.P. would be \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}\).
• Unlike A.P. or G.P., there is no standard formula for the nth term or the sum of an H.P.

Understanding H.P. is helpful to set it apart from the other types of progressions, particularly when exploring solutions to exercises like the one given, where different types of progressions need to be identified. In our exercise, since the sums formed a geometric progression, it highlighted that H.P. was not applicable, serving as a distinct contrast to what was needed.