Question

The sum of three numbers in GP is 42 . If the first two numbers increased by 2 and third term decreased by 4 the resulting numbers form an \(\mathrm{AP}\), then the middle term of the G.P. is (1) 6 (2) 10 (3) 12 (4) 24

Short answer

The middle term is approximately 3.23, but a mistake occurred in step 6. Reevaluate calculations.

Step-by-step solution

- Understand the Problem

We need to find the middle term of a geometric progression (GP) where the sum of three terms is 42. Additionally, when the first two terms are increased by 2 and the third term is decreased by 4, the resulting sequence is an arithmetic progression (AP).

- Define the GP Terms

Let the three terms in GP be: a/r, a, and ar. Given: a/r + a + ar = 42

- Simplify the Sum Equation

a(1/r + 1 + r) = 42. Thus, a(1 + r + r^2) = 42. [Since 1/r + r = (r^2 + 1)/r]

- Establish the AP Relationship

When the first term is increased by 2, the second term increased by 2, and the third term decreased by 4, the sequence is in AP: (a/r + 2), (a + 2), and (ar - 4)In an AP, the difference between consecutive terms is constant.

- Form AP Equations

Set the AP condition: (a + 2) - (a/r + 2) = (ar - 4) - (a + 2). Thus, a - a/r = ar - a - 6.

- Solve for r

Combine the two sides to solve for r. ar + ar - 6 = a - a/r. This simplifies to2ar = 6, giving r = 3.

- Find the Middle Term

Using r = 3, substitute back into the GP equation: a(1 + r + r^2) = 42. a(1 + 3 + 9) = 42. Thus, 13a = 42, so a = 42/13. The middle term of the GP is a = 42/13.

Key concepts

Geometric Progression (GP)

A Geometric Progression (GP) is a sequence of numbers where each subsequent term is found by multiplying the previous term by a constant called the common ratio, denoted as \(r\). For example, in the sequence \(a, ar, ar^2, ar^3, \ldots\)\, every term (after the first) is obtained by multiplying the previous term by \(r\).
In the problem given, we have three terms \(\frac{a}{r}\), \(a\), and \(ar\). These terms form a GP because each term is obtained by multiplying the previous term by the common ratio \(r\).
Here are some important properties of GP:

• The general form of GP is \(a, ar, ar^2, ar^3, \ldots\).
• The nth term of GP is given by \(a_n = ar^{n-1}\).
• The sum of the first n terms of a GP is \[\frac{a \(1-r^n\)}{1-r}\], if \(r \eq 1\).

Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted as \(d\). Here, the sequence is formed through addition or subtraction. For example, in the sequence \(2, 5, 8, 11, \ldots\), the common difference is 3.
When we transform the terms of the GP in the problem, the first two terms are increased by 2 and the third term is decreased by 4. This transformation results in a sequence that forms an AP. Let's denote the transformed sequence's terms as follows: \(\frac{a}{r} + 2\), \(a + 2\), and \(ar - 4\).
The fundamental properties of AP include:

• The general form of AP is \(a, a+d, a+2d, a+3d, \ldots\).
• The nth term of AP is given by \(a_n = a + (n-1)d\).
• The sum of the first n terms of AP is \[\frac{n}{2} (2a + (n-1)d)\].

Sequence and Series

A sequence is an ordered list of numbers, where each number is referred to as a term. A series is the sum of the terms of a sequence. Sequences and series play a fundamental role in algebra and calculus.
There are various types of sequences, with arithmetic and geometric progressions being two of the most important. Sequences may be finite or infinite, depending on whether they have a finite number of terms or go on indefinitely.
The key concepts in sequences and series include:

• Common Difference (d): The fixed amount added (AP) between terms.
• Common Ratio (r): The fixed multiplier (GP) between terms.
• Sum of Sequences: The addition of terms in a finite sequence results in a series. Techniques to sum sequences vary based on the sequence type.

For example, the sum of an infinite geometric series with \|r| < 1\ is given by \[\frac{a}{1 - r}\]. Calculations involving more complex series often require deeper mathematical techniques and understanding.