Problem 77 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The sum of the \(3^{\text {rd }}\) and the \(4^{\text {th }}\) terms of a GP. is 60 and the product of its first three terms is 1000 . If the first term of this G.P. is positive, then its \(7^{\text {th }}\) term is : [Online April 11, 2015] (a) 7290 (b) 640 (c) 2430 (d) 320

Step-by-Step Solution

Answer

The 7th term of the G.P. is 320.

Step 1: Understanding the given information

We are given a geometric progression (G.P.) where the sum of the 3rd and 4th terms is 60, and the product of the first three terms is 1000. Our task is to find the 7th term of this G.P. with the additional condition that the first term of the G.P. is positive.

Step 2: Setting up expressions for the G.P. terms

Let the first term of the G.P. be \( a \) and the common ratio be \( r \). The terms of the G.P. are:- 1st term: \( a \)- 2nd term: \( ar \)- 3rd term: \( ar^2 \)- 4th term: \( ar^3 \)

Step 3: Formulating equations from given conditions

We have the equation for the sum of the 3rd and 4th terms: \( ar^2 + ar^3 = 60 \). Equation (1)And for the product of the first three terms: \( a \times ar \times ar^2 = a^3r^3 = 1000 \). Equation (2)

Step 4: Solving for the first equation

Factor the sum: \( ar^2 (1 + r) = 60 \) From Equation (1).

Step 5: Solving for common ratio from equation 2

From Equation (2), solve for \( a^3r^3 = 1000 \), we get:\( a^3r^3 = (10)^3 \), so \( a^3r^3 = 10^3 \)This implies \( ar = 10 \).

Step 6: Solving for specific terms and verifying

Substitute \( ar = 10 \) into Equation (1):\( 10r(1 + r) = 60 \)\( 10 + 10r = 60 \) \( 10r^2 + 10r - 60 = 0 \)Divide throughout by 10:\( r^2 + r - 6 = 0 \) solve by factorization:\((r-2)(r+3) = 0 \) \( r = 2 \text{ or } r = -3 \)But since the first term is positive and we want 7th term to be positive, choose \( r= 2 \).

Step 7: Calculating the 7th term

Using the formula 7th term \( a r^6 \). Substitute \( a = \frac{10}{r} = \frac{10}{2} = 5 \):\( 7^{\text{th}} \text{ term } = 5 \times 2^6 = 5 \times 64 = 320 \).

Key Concepts

Sequence

A **sequence** is an ordered list of numbers where each number is called a term. In a geometric progression (G.P.), the sequence is organized in such a way that each term after the first is found by multiplying the previous term by a constant called the common ratio. Here is how a G.P. works:
\( a, ar, ar^2, ar^3, \ldots \)
In this sequence, \( a \) is the first term and \( r \) is the common ratio. Each term is derived by multiplying the previous term by the common ratio. Understanding this pattern is crucial. Once the first term and the common ratio are known, any term in the sequence can be calculated using the formula \( a r^{(n-1)} \), where \( n \) is the term number in the sequence.
Remember, in the context of a G.P., knowing the order helps determine its properties. It's the foundation for solving problems related to geometric sequences.

Common Ratio

The **common ratio** is a key element in a geometric progression, differentiating it from other types of sequences. It’s what sets a G.P. apart by indicating how the sequence progresses from one term to the next. In a G.P., every term is a result of multiplying the previous term by the common ratio \( r \). For example, if the common ratio is 2, each term is twice the previous term.
Let's see how it works with an example:

• First term: \( a \)
• Second term: \( ar \)
• Third term: \( ar^2 \)
• Fourth term: \( ar^3 \)

Understanding how to find and use the common ratio is essential because it allows you to predict any term in your sequence. If you know one term and the common ratio, you can compute the others.
In many problems, you may have to find \( r \) from the properties of the sequence, such as in product or sum conditions, as seen in the exercise where equations are formed using known properties.

Terms

In a geometric progression, each number in the sequence is known as a "term." Each term is pivotal in understanding and solving problems related to G.P. To identify or calculate a specific term, you typically use a set formula that involves the common ratio and the number of terms in the sequence.
For a specific term in a G.P., the formula used is:
\( n^{ ext{th}} \text{ term} = a \, r^{n-1} \)
For instance, in the exercise, the **7th term** is what we're searching for, which is calculated as \( ar^6 \). It's important to note the progression of each term in context to the entire sequence:

• If the first term \( a \) is known, and common ratio \( r \) is found or given, determining any specific term just involves substituting \( n \) in the formula.
• Each term has a unique position and value, shaping how the sequence behaves overall.

This structured pattern in terms of a G.P. empowers you to flexibly and accurately determine any part of the sequence.