Problem 133 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For a sequence \(\left\\{a_{n}\right\\}, a_{1}=2\) and \(\frac{a_{n+1}}{a_{n}}=\frac{1}{3}\). Then find the value of \(\sum_{r=1}^{20} a_{r}\).

Step-by-Step Solution

Answer

The sum of the first 20 terms of the sequence is approximately 3.

Step 1: Determine the common ratio

As given in the prompt, we know that: \(\frac{a_{n+1}}{a_n} = \frac{1}{3}\) Since the ratio between consecutive terms is constant, we can say there is a common ratio, r = \(\frac{1}{3}\), that characterizes this as a geometric sequence.

Step 2: Find the formula for the general term

In a geometric sequence, the general term \(a_n\) is given by the formula: \(a_n = a_1 \times r^{n-1}\) We are given \(a_1 = 2\) and we found the common ratio r = \(\frac{1}{3}\). Plugging these values into the formula, we get: \(a_n = 2 \times \left(\frac{1}{3}\right)^{n-1}\)

Step 3: Apply the sum formula for geometric series

The sum of the first n terms of a geometric series, \(S_n\) is given by the formula: \(S_n = a_1 \times \frac{1 - r^n}{1 - r}\) Now, we want the sum of the first 20 terms, so n = 20. Plugging in the known values, we have: \(S_{20} = 2 \times \frac{1 - \left(\frac{1}{3}\right)^{20}}{1 - \frac{1}{3}}\)

Step 4: Compute the sum

Simplify the expression for \(S_{20}\) to find the sum: \(S_{20} = 2 \times \frac{1 - \left(\frac{1}{3}\right)^{20}}{\frac{2}{3}}\) \(S_{20} = 3 \times \left(1 - \left(\frac{1}{3}\right)^{20}\right)\) Now, calculate the value: \(S_{20} = 3 \times \left(1 - \frac{1}{3486784401}\right) \approx 2.999999999 \) The sum of the first 20 terms of the sequence is approximately 3.

Key Concepts

Common Ratio

The common ratio is a crucial concept in geometric sequences. It's the factor by which we multiply one term to get the next term in the sequence. In a geometric sequence, this ratio remains consistent. For example, if the common ratio is \( \frac{1}{3} \), as in our exercise, each term is \( \frac{1}{3} \) of the previous term.

Understanding the common ratio is essential because it determines the nature and behavior of the sequence:

• It defines the direction of the sequence (increasing or decreasing).
• It's used in formulas to find any term within the sequence.

To identify the common ratio, you can divide any term by the previous term. For example, if \( a_2 = \frac{2}{3} \) and \( a_1 = 2 \), then the common ratio \( r \) is \( \frac{a_2}{a_1} = \frac{\frac{2}{3}}{2} = \frac{1}{3} \). Recognizing the common ratio allows us to apply other geometric formulas effectively.

General Term of Geometric Sequence

The general term of a geometric sequence is a formula that enables us to calculate the nth term without listing all preceding terms. It's like a shortcut! The formula for the nth term is:\[ a_n = a_1 \times r^{n-1} \]

In this formula:

• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number we want to find.

Let's break it down:

• For our exercise, where \( a_1 = 2 \) and \( r = \frac{1}{3} \), this means \( a_n = 2 \times \left(\frac{1}{3}\right)^{n-1} \).
• If we want to find the 5th term, plug in \( n = 5 \) to get \( a_5 = 2 \times \left(\frac{1}{3}\right)^4 \).

This approach simplifies finding any term in the sequence, especially when dealing with large sequences, because you don't have to compute each term sequentially.

Sum Formula for Geometric Series

The sum formula for a geometric series is a powerful tool for finding the total of a set number of terms in a geometric sequence. The formula is:\[ S_n = a_1 \times \frac{1 - r^n}{1 - r} \]

Here's how it works:

• \( S_n \) represents the sum of the first n terms.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the number of terms we're summing.

For example, in the exercise:

• With \( a_1 = 2 \), \( r = \frac{1}{3} \), and \( n = 20 \), the sum becomes \( S_{20} = 2 \times \frac{1 - \left(\frac{1}{3}\right)^{20}}{1 - \frac{1}{3}} \).
• This formula lets you find the sum of terms quickly without adding each one individually.

Using the formula is particularly handy in cases where the sequence involves many terms, as it reduces the problem to a few simple calculations. The sum formula reveals that regardless of how long the sequence is, you can always predict the total value by understanding the first term and the common ratio!