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

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: Understanding the Problem

You are given the mean of \(n\) items, \(\bar{x}\), and each item is increased by a power of 3: \(3,3^2,3^3, \ldots, 3^n\). You need to find the new mean after all items are increased by these amounts.

Step 2: Calculate the Total Increase

To calculate the total increase, you sum up the additional amounts added to each item: \(3 + 3^2 + 3^3 + \ldots + 3^n\). This sum is a geometric series with the first term \(a = 3\) and the common ratio \(r = 3\).

Step 3: Sum of the Geometric Series

The formula for the sum of a geometric series \(S_n\) with \(n\) terms, first term \(a\), and common ratio \(r\) is \(S_n = a\frac{r^n - 1}{r - 1}\). For our series, \(S_n = 3\frac{3^n - 1}{3 - 1} = \frac{3}{2}(3^n - 1)\).

Step 4: New Total Sum of All Items

If the original sum of all \(n\) items was \(n\bar{x}\), the new sum after each item's increase is \(n\bar{x} + \frac{3}{2}(3^n - 1)\).

Step 5: Calculate the New Mean

The new mean is the new total sum divided by the number of items: \[\bar{x}_{new} = \frac{n\bar{x} + \frac{3}{2}(3^n - 1)}{n} = \bar{x} + \frac{3(3^n - 1)}{2n}.\]

Step 6: Compare to Answer Choices

Compare the new mean expression \(\bar{x} + \frac{3(3^n - 1)}{2n}\) with the given options. It matches Option (B): \(\bar{x} + 3 \frac{(3^n - 1)}{2n}\).

Key Concepts

Mean of a Sequence

The mean, also known as the average, is a measure of central tendency that helps us understand the typical value within a collection of numbers. It is calculated by summing up all the elements in the sequence and dividing by the number of elements.To find the mean of a sequence, follow these simple steps:

• Add up all the numbers in the sequence.
• Count the total number of elements.
• Divide the sum by the number of elements.

For example, if you have a sequence of numbers like \(2, 4, 6\), you would find the sum \(2 + 4 + 6 = 12\), and then divide by the number of elements \((3)\).So, the mean would be \( \frac{12}{3} = 4\).This simple process helps to identify the central value of the set of data. In the context of the problem, the mean \(\bar{x}\) represents the initial average value of the \(n\) items before any changes are made.

Summation of Series

A geometric series is a series of numbers in which each term after the first is found by multiplying the previous term by a constant, known as the common ratio.The geometric series in this problem is defined as \(3, 3^2, 3^3, \ldots, 3^n\), where each term is a power of 3.To find the total sum of a geometric series, we use the sum formula:\[S_n = a \frac{r^n - 1}{r - 1},\]where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.In our case, the series starts with 3, and the common ratio is also 3. So the formula becomes:\[S_n = 3 \frac{3^n - 1}{3 - 1} = \frac{3(3^n - 1)}{2}.\] This sum gives us the total increase in the value of all items after the series is added to them.Understanding the summation of a geometric series is key to solving problems where each element increases exponentially.

Increased Sequence Mean

When each item in a sequence is increased by a certain amount, it impacts the overall mean of the sequence. First, let's look at how these increases affect the sum of the sequence:- Originally, all items sum up to \(n\bar{x}\), where \(n\) is the number of items and \(\bar{x}\) is the mean.- The additional total from increasing each element by powers of 3 is given by the geometric series sum \(\frac{3(3^n - 1)}{2}\).
To find the new mean, divide the updated total sum by \(n\):\[\bar{x}_{new} = \frac{n\bar{x} + \frac{3(3^n - 1)}{2}}{n}.\]Thus, the new mean becomes:\[\bar{x}_{new} = \bar{x} + \frac{3(3^n - 1)}{2n}.\]This shows that each increase in value shifts the central or average value to the right by this calculated amount.Understanding how changes affect the mean offers insight into the dynamics of data sets in mathematical analyses and real-world scenarios.