A teacher gave a sum to his class to find the average of 'n' number wiz 1, 2, 3, ... n. But when the teacher checked the solution, he found that during calculation, a student just missed a number for addition and thus his average of 'n' numbers was 15. The least value of n isa)n = 29b)n = 30c)n = 31d)n = 32Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Let the missing number be k.

The sum of series 1, 2, 3, …..n is n(n+1)/2

So, according to problem statement

n2 + n – 2k = 30n

n2 – 29n = 2k

n (n – 29) = 2k

So, at n ≤ 29, the above expression is invalid. Because at n ≤ 29, the value of k become either zero or negative.

So, least value of n is 30.

At n = 30, we have

30(30 – 29) = 2k

k = 15

Answer

The problem states that a teacher gave a sum to his class to find the average of 'n' numbers, starting from 1 to n. However, one student missed a number during the calculation, resulting in an average of 15 for the 'n' numbers. We need to determine the least value of 'n' that satisfies this condition.

To solve this problem, we can use the formula for the sum of an arithmetic series:

Sn = (n/2)(a + l)

Where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.

Let's break down the problem step by step:

1. Calculate the sum of the 'n' numbers without the missing term:

Sn = (n/2)(1 + n) - x

Where x is the missing term.

2. Calculate the sum of the 'n' numbers using the average:

Sn = 15n

3. Equate the two equations:

(n/2)(1 + n) - x = 15n

4. Simplify and rearrange the equation:

(n^2 + n)/2 - x = 15n

n^2 + n - 2x = 30n

n^2 - 29n + 2x = 0

5. Use the quadratic formula to solve for 'n':

n = (-b ± √(b^2 - 4ac))/(2a)

In this case, a = 1, b = -29, and c = 2x.

6. Since we are looking for the least value of 'n', we can consider only the positive root of the quadratic equation:

n = (-(-29) + √((-29)^2 - 4(1)(2x)))/(2(1))

n = (29 + √(841 - 8x))/2

7. The value of 'n' should be an integer, so we need to find the least value of 'x' that makes the expression inside the square root a perfect square.

8. By trial and error, we can determine that 'x' should be 6 to obtain a perfect square:

n = (29 + √(841 - 8(6)))/2

n = (29 + √(793))/2

n ≈ 29.82

9. Since 'n' should be an integer, the least value of 'n' that satisfies the condition is 30 (Option B).

Therefore, the correct answer is option 'B' - n = 30.

Similar Mechanical Engineering Doubts

In an exam, the average was found to be 50 marks. After deducting computational errors the marks of the 100 candidates had to be changed from 90 to 60 each and average came down to 45 marks. Total number of candidates who took the exam was

The average Nusselt number in laminar natural convection from avertical wall at 180C with still air at 20C is found to be 48. If the wall temperature becomes 30C, all other parameters remaining same, the average Nusselt number will be

A 50-Kg block of iron casting at 500K is thrown into a large lake that is at a temperature of 285K. The iron block eventually reaches thermal equilibrium with the lake water. Assuming an average specific heat of 0.45 kJ/Kg-K for the iron, the entropy generation during theprocess is

The system of linear equations 4x + 2y = 7 2x + y = 6 has

Top Courses for Mechanical EngineeringView all

General Aptitude for GATE

Engineering Mathematics

General Awareness & Knowledge

GATE Mechanical (ME) Mock Test Series 2025

Strength of Materials (SOM)

Top Courses for Mechanical Engineering

Top Courses for Mechanical Engineering

Related Content

Mock Test Series

Subject-wise Courses

Subject-wise Courses

Other Engg Branches

Company