Problem 56 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(a_{n}\) be the \(n^{\text {th }}\) term of a G.P. of positive terms. If \(\sum_{n=1}^{100} a_{2 n+1}=200\) and \(\sum_{n=1}^{100} a_{2 n}=100\), then \(\sum_{n=1}^{200} a_{n}\) is equal to: [Jan. 9, 2020 (II)] (a) 300 (b) 225 (c) 175 (d) 150

Step-by-Step Solution

Answer

The sum \(\sum_{n=1}^{200} a_{n}\) equals 300. Answer: (a) 300.

Step 1: Understand Given Terms

We are given two separate series sums involving the terms of a geometric progression (G.P.). The summation \(\sum_{n=1}^{100} a_{2n+1}=200\) represents the sum of 100 terms where each term is of the form \(a_{2n+1}\). On the other hand, the summation \(\sum_{n=1}^{100} a_{2n}=100\) represents the sum of 100 terms where each term is of the form \(a_{2n}\).

Step 2: Express Summations in G.P. Form

For a geometric progression, the general term can be represented using the first term \(a\) and the common ratio \(r\) as \(a_n = a \cdot r^{n-1}\). Therefore, \(a_{2n+1} = a \cdot r^{2n}\) and \(a_{2n} = a \cdot r^{2n-1}\).

Step 3: Calculate Individual Sums

The sum \(\sum_{n=1}^{100} a_{2n+1}=200\) can be expressed as a geometric series: \[ a \cdot r^2 + a \cdot r^4 + \ldots + a \cdot r^{200} = 200 \]Similarly, the sum \(\sum_{n=1}^{100} a_{2n}=100\) can be expressed as: \[ (a \cdot r) + (a \cdot r^3) + \ldots + (a \cdot r^{199}) = 100 \]

Step 4: Relationship Between Series

Notice that each term of the second sum is \(r\) times each term of the first sum, i.e., the terms in \(S_2\) are the terms in \(S_1\) multiplied by \(r\). Thus, the sums relate by the equation: \[ S_2 = r \cdot S_1 \]Substituting known values: \[ 100 = r \cdot 200 \]This gives \(r=0.5\).

Step 5: Find Total Sum

Now, we need to calculate \(\sum_{n=1}^{200} a_{n}\), which is the sum of all terms in the progression. We utilize the first sum (\(S_1\)) and second sum (\(S_2\)): \[ S = S_1 + S_2 = 200 + 100 = 300 \]

Step 6: Conclusion

The total sum \(\sum_{n=1}^{200} a_{n}\) is calculated by adding the results from the summations of odd and even indexed terms.

Key Concepts

Sequences and Series

In mathematics, a sequence refers to a set of numbers arranged in a specific order following a particular rule. A series is the sum of the terms of a sequence. Sequences are foundational in understanding mathematics and play a crucial role in higher-level studies, such as in calculus and discrete mathematics. Common sequences include arithmetic and geometric sequences. The sum of an infinite sequence can also converge to a particular value, depending on the properties of the sequence. Understanding sequences and their related series help in solving complex mathematical problems by breaking them down into simpler, more manageable parts.

• Arithmetic Sequences: Have a constant difference between terms.
• Geometric Sequences: Each term is a constant multiple of the previous term.

It’s important to distinguish between the different types of sequences and series as this helps in the correct application of formulas and solving methods in a range of mathematical problems.

Geometric Progression (G.P.)

A geometric progression (G.P.) is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. Geometric progressions are widely used in various branches of mathematics as they can model exponential growth or decay, relevant in fields like finance, sciences, and engineering. In the general form of a G.P., each term can be expressed as:

• General Term: If the first term is denoted by \( a \) and the common ratio by \( r \), then the \( n^{th} \) term is given by \( a_n = a \, r^{n-1} \).

Understanding the concept of a geometric progression allows for calculation of specific terms or sums within the sequence. For example, by knowing just the first term and common ratio, you can easily calculate any term in the sequence or find the sum of several terms.

Summation of Series

Summing a series refers to finding the total when all terms in the sequence are added together. For a geometric series, this involves adding all the terms from a geometric progression. The formula to find the sum of the first \( n \) terms of a G.P. is:

• For \( n \) terms: \( S_n = a \frac{1 - r^n}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio ( \( r eq 1 \)).

This formula allows us to quickly determine the sum without needing to add each individual term, which is especially useful for large series. In cases where there are infinite terms and the common ratio \( |r| < 1 \), the series converges, and the sum is \( \frac{a}{1 - r} \). Being able to sum series is crucial in advanced mathematics and practical applications, like calculating interest in finance.

Common Ratio

The common ratio in a geometric progression is the factor by which we multiply a term to get the next term. It is a vital component of the sequence as it dictates the progression's behavior, whether it grows or shrinks. To determine the common ratio, divide any term by its preceding term:

• Formula: \( r = \frac{a_{n+1}}{a_n} \)

Understanding and calculating the common ratio is crucial as it allows you to predict the pattern and next terms of the sequence. It also plays a key role in determining the series’ sum, as seen in the exercise involving the equations \( S_2 = r \cdot S_1 \). A common ratio less than 1 indicates diminishing terms, while greater than 1 indicates the terms are growing.