Problem 29 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The mean of \(n\) items is \(\bar{x}\). If each item is successively increased by \(3,3^{2}, 3^{3}, \ldots .3^{n}\), then new mean equals (A) \(\bar{x}+\frac{3^{n+1}}{n}\) (B) \(\bar{x}+3 \frac{\left(3^{n}-1\right)}{2 n}\) (C) \(\bar{x}+\frac{3^{n}}{n}\) (D) \(\bar{x}+3 \frac{\left(3^{n}-1\right)}{2 n}\)

Step-by-Step Solution

Answer

The new mean is option (B): \(\bar{x}+3 \frac{(3^{n}-1)}{2n}\).

Step 1: Calculate Original Total Sum

The original mean of the items is given as \( \bar{x} \). Therefore, the total sum of the \( n \) items is \( n \times \bar{x} \).

Step 2: Calculate Change in Each Item

Each item is successively increased by powers of 3. This means the increments are \( 3, 3^2, 3^3, \ldots, 3^n \).

Step 3: Calculate Total Change in Value

Sum the series of changes: \( 3 + 3^2 + 3^3 + \ldots + 3^n \). This is a geometric series with the first term \( a = 3 \) and common ratio \( r = 3 \), up to \( n \) terms.

Step 4: Sum of Geometric Series Formula

Use the formula for the sum of a geometric series: \( S_n = a \frac{r^n - 1}{r - 1} \). Plugging in our values gives: \( 3 \frac{3^n - 1}{3 - 1} = \frac{3(3^n - 1)}{2} \).

Step 5: Calculate New Total Sum

Add the total change to the original total sum: \( n \times \bar{x} + \frac{3(3^n - 1)}{2} \).

Step 6: Calculate New Mean

Divide the new total sum by \( n \) to find the new mean: \( \bar{x} + \frac{3(3^n - 1)}{2n} \).

Step 7: Select the Correct Option

Comparing this new mean with the options, the correct one is (B): \( \bar{x} + 3 \frac{\left(3^{n}-1\right)}{2 n} \).

Key Concepts

Understanding Mean Calculation

The mean, often referred to as the average, is a central concept in statistics that helps summarize a set of numbers. It's calculated by adding up all the individual values and then dividing by the number of items you have. This gives you one single number that represents the central tendency of the data.
To better understand how this is applied in practical situations, consider that the mean is not just about numbers in isolation. Instead, it tells a story about all the data points at once. For example, if you have three test scores: 85, 90, and 95, the mean would be calculated as follows:

• Add together the numbers: 85 + 90 + 95 = 270
• Divide the total by the number of scores: 270 / 3 = 90

The mean score is 90, which acts as a benchmark for the typical performance across your tests. This is particularly useful for understanding how adjustments can affect overall performance in incremental changes or series.

Increment in Sequence

Increment refers to a gradual increase in quantity. In sequences, increments often follow a specific pattern. In mathematical sequences, a common form of increment is represented through constant additions or multiplications.
In this exercise, each item in the original set is being increased by powers of 3. This is a special case of increment represented by:

• First increment by 3
• Second increment by 9 which is 3 squared
• Third increment by 27 which is 3 cubed, and so on

Such patterns are crucial because they help us predict future terms and understand how totals change over time. Recognizing these patterns also allows for more efficient calculation without the need to add each increment individually each time.

Geometric Progression

Geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the exercise, increments of 3, 9, 27, etc., form a geometric sequence. The first term is 3 and the common ratio also is 3.
Such sequences are characterized by their exponential growth or decay, depending on the common ratio.

• Common ratio greater than one: sequence grows exponentially.
• Common ratio between zero and one: sequence shrinks.

Understanding geometric progression is fundamental in calculating sums of series like the problem above, where it helps efficiently compute total increases.

Sum of Powers

The sum of powers in a geometric series is a concept that allows us to add up a sequence of numbers that are raised to incrementing powers. Specifically, when confronted with series like 3, 9, 27, ..., these numbers are powers of 3.
To calculate the sum of such a series where each term is found by multiplying by a consistent base, we use the formula to find the sum of a geometric series:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Here, for example, the first term \( a \) is 3, the ratio \( r \) is also 3, and you determine \( n \) depending on how many terms you have. Using our knowledge of powers and geometric series formulae lets us find the total sum quickly, which could otherwise be computationally intensive.