Problem 112 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The sum of of first ten terms of an A.P. is equal to 155 and the sum of first two terms of a G.P. is \(9 .\) If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to the common difference of the A.P, then (A) first term of the G.P. is \(\frac{2}{3}, 3\) (B) first term of the A.P. is \(\frac{2}{3}, 3\) (C) Common ratio of the G.P. is \(\frac{25}{2}, 2\) (D) Common difference of the A.P is \(\frac{2}{3}, 3\)

Step-by-Step Solution

Answer

Options A and B are correct.

Step 1: Understand the problem statements

We need to find the terms and differences of an arithmetic progression (A.P.) and a geometric progression (G.P.) based on given conditions. The sum of the first 10 terms of the A.P. is 155, and the sum of the first two terms of the G.P. is 9. The first term of the A.P. equals the common ratio of the G.P., and the first term of the G.P. equals the common difference of the A.P.

Step 2: Define variables

Let the first term of the A.P. be \(a\) and the common difference be \(d\). Then the sum of the first 10 terms \(S_{10}\) is given by the formula: \[ S_{10} = \frac{10}{2} \times (2a + 9d) = 155 \]

Step 3: Solve for A.P. terms

Solving for \(2a + 9d\), we get: \[ 5(2a + 9d) = 155 \] \[ 2a + 9d = 31 \] (Equation 1)

Step 4: Examine the G.P. conditions

Let the first term of the G.P. be \(b\) and the common ratio be \(r\). We know that the sum of the first two terms is 9: \[ b + br = 9 \] \[ b(1 + r) = 9 \]

Step 5: Use the relationship between A.P. and G.P.

We have the relationships: - First term of A.P. \(a = r\) (the common ratio of the G.P)- First term of G.P. \(b = d\) (the common difference of the A.P)

Step 6: Substitute relationships into equations

Substitute \(a = r\) and \(d = b\) into Equation 1: \[ 2r + 9b = 31 \] (Equation 2)And \(b(1 + r) = 9\) from Step 4.

Step 7: Solve the system of equations

Using the system of equations:1. \( 2r + 9b = 31 \)2. \( b(1 + r) = 9 \)From equation 2, solve for \(b\):\[ b = \frac{9}{1+r} \]Substitute into Equation 1:\[ 2r + 9\left(\frac{9}{1+r}\right) = 31 \] Solve to find two sets of solutions: \((r, b) = (3, \frac{2}{3})\) and \((r, b) = (\frac{2}{3}, 3)\).

Step 8: Check possible solutions against options

These solutions directly indicate that:- First term of the A.P. is either \(3\) or \( \frac{2}{3} \)- First term of the G.P. is either \( \frac{2}{3} \) or \(3\)So the correct options are:(A) first term of the G.P. is \( \frac{2}{3}, 3\)(B) first term of the A.P. is \( \frac{2}{3}, 3\)

Key Concepts

Geometric Progression

A geometric progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. This progression is recognized by the multiplication factor applied to each term.
An example of a G.P. could be the series 2, 4, 8, 16,... where each term is multiplied by the common ratio of 2. In the exercise we are discussing, the first two terms of a G.P. add up to 9, giving rise to a system of equations that helps to determine the common ratio and first term.

• Understanding the basic structure of G.P. is crucial to tackling any question involving sequences.
• The first term and the common ratio are necessary to define a geometric progression.

With clarity on the concept of geometric progression, solving problems involving these series becomes much more intuitive.

Sum of Series

The sum of a series refers to the total of all terms added together in the sequence. For both arithmetic and geometric series, calculating the sum is a fundamental concept:

• In an arithmetic progression (A.P.), the sum of the first few terms can be calculated using the formula: \[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
• For a geometric progression (G.P.), the sum can be calculated using: \[ S_n = a \frac{(r^n - 1)}{r - 1} \] \text{for } r eq 1

In the given exercise, the sum of the first 10 terms of an A.P. is used to find connections between the series that help solve the problem. Understanding how to calculate these sums provides a solid foundation for exploring the relationships within the sequences.

Common Ratio

The common ratio in a geometric progression is the constant factor between consecutive terms. It is crucial, as it defines how the sequence proceeds by multiplying the previous term by this ratio to get the next term. For instance, if the first term is 3 and the common ratio is 2, the series progresses as 3, 6, 12, 24, and so on.
In the context of our exercise, knowing the common ratio allows us to form and solve equations to find the necessary sequence values. Here's how:

• Write the general term of the G.P. as \( ar^{n-1} \) for the nth term.
• Use the formula for the sum of terms to incorporate the common ratio.
• Apply the given condition, such as the common ratio equaling the first term of the A.P., to derive further relationships.

Grasping the role of the common ratio is essential for decoding and constructing geometric progressions.

Common Difference

The common difference is a defining characteristic of an arithmetic progression (A.P.), representing the consistent interval between consecutive terms. It is the value you add to any term to get to the next one in the series. For example, with a common difference of 4, a series starting at 1 would progress as 1, 5, 9, 13, etc.
In this problem, the common difference is linked to the first term of the G.P., creating interconnected relationships between the series. This linkage can be explored as follows:

• Understand that each term in the A.P. is represented as \( a + (n-1)d \).
• For the problem solution, solve equations where the common difference plays a central role in calculating the sums or deriving other properties of the series.

Appreciating the common difference offers insight into the structure of arithmetic sequences and primes you towards solving related problems effectively.