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Question

The arithmetic mean of \(n\) numbers is \(M\). If the sum of first \((n - 1)\) terms is \(k\), then what is the \(n\)th number?

This question was previously asked in
CDS 2 2024 Maths Question Paper (01-Sep-2024)
The correct answer is
\(nM-k\)

Calculating the Nth Number Using Arithmetic Mean

This solution explains how to find the value of the nth number when the arithmetic mean of \(n\) numbers is given, along with the sum of the first \((n-1)\) numbers.

Understanding Arithmetic Mean

The arithmetic mean (or average) of a set of numbers is calculated by dividing the sum of all the numbers by the count of the numbers in the set.

Mathematically, if we have \(n\) numbers, let them be \(x_1, x_2, ..., x_n\). The arithmetic mean (\(M\)) is given by the formula:

\( M = \frac{x_1 + x_2 + ... + x_n}{n} = \frac{\sum_{i=1}^{n} x_i}{n} \)

Applying the Formula to the Problem

We are given:

  • The total count of numbers is \(n\).
  • The arithmetic mean of these \(n\) numbers is \(M\).
  • The sum of the first \((n - 1)\) numbers is \(k\). Let this sum be \(S_{n-1} = k\).
  • We need to find the \(n\)th number, let's call it \(x_n\).

Step-by-Step Solution

  1. Calculate the total sum of the \(n\) numbers: From the definition of the arithmetic mean, we know that:

    \( M = \frac{\text{Sum of } n \text{ numbers}}{n} \)

    Rearranging this formula to find the sum of all \(n\) numbers (let's call it \(S_n\)):

    \( S_n = M \times n \)

    So, the total sum of the \(n\) numbers is \(nM\).

  2. Relate the total sum to the given sum: The total sum (\(S_n\)) is the sum of the first \((n - 1)\) numbers plus the \(n\)th number (\(x_n\)).

    \( S_n = (x_1 + x_2 + ... + x_{n-1}) + x_n \)

    We know that the sum of the first \((n - 1)\) numbers is \(k\). So:

    \( S_n = k + x_n \)

  3. Solve for the \(n\)th number (\(x_n\)): Now we have two expressions for the total sum \(S_n\):
    • \(S_n = nM\)
    • \(S_n = k + x_n\)

    By setting these two expressions equal to each other:

    \( nM = k + x_n \)

    To find the \(n\)th number (\(x_n\)), we isolate it by subtracting \(k\) from both sides of the equation:

    \( x_n = nM - k \)

Conclusion

Therefore, the value of the nth number is \(nM - k\). This is derived directly from the definition of the arithmetic mean and the information provided about the sum of the first \((n-1)\) terms.

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Similar Questions

  1. A person purchases one kg of tea powder from each of the four places \(A, B, C, D\) at the rate of ₹1000 per 1 kg, 2 kg, 4 kg, 5 kg. If on an average he purchased \(x\) kg of tea powder per ₹1000, then what is the approximate value of \(x\) ?

  2. The sum of two positive numbers is 40. If the GM of these two numbers is lower than their AM by 20%, then what is the difference between the two numbers?
  3. The average of the temperatures recorded at noontime from Monday to Sunday is \(31^\circ C\). If the lowest temperature recorded is \(30^\circ C\), then what is the maximum of temperature that is possible to record at noontime on any one of the days
  4. What is the difference between the average of first 50 even natural numbers and the average of first 50 odd natural numbers ?
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  6. All possible groups of 3 distinct numbers from among A, B, C, D and E are formed. If the aggregate of sums of numbers of each group is 120, then what is the arithmetic mean of A, B, C, D and E ?
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Important Questions from Average

  1. Average of 40 numbers is 71, if the number 100 replaced by 140, then average is increased by

  2. The captain of a football team of 11 members is 28 years old and the goalkeeper is 4 years older than him. If the ages of these two are removed, then the average age of the remaining players is two years less than the average age of the whole team. What is the average age of the team?

  3. There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B.

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    1. The average score of Class-B will definitely decrease.

    2. The average score of Class-A will definitely increase.

    Which of the above statements is/are correct?

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  5. A cow costs more than 4 goats but less than 5 goats. If a goat costs between Rs. 600 and Rs. 800, which of the following is a most valid conclusion?

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