Natural number MCQ Quiz - Objective Question with Answer for Natural number - Download Free PDF

Given:

First 48 natural numbers.

Concept used:

Sum of squares of first n natural Number = [n (n + 1) (2n + 1)]/6

Average of sum of squares of first n natural Number = [(n + 1) (2n + 1)]/6

Calculation:

Sum of squares of first 48 natural Number = [(48 + 1)(2 × 48 + 1)]/6 = 792.17

∴ The average of squares of first 48 natural numbers is 792.17

Given:

First 125 natural numbers.

Formula Used:

Average of first n natural numbers = (n + 1) / 2

Calculation:

n = 125

Average = (125 + 1) / 2

⇒ Average = 126 / 2

⇒ Average = 63

The average of the first 125 natural numbers is 63.

Given:

The first 47 natural numbers.

Formula used:

The sum of squares of the first n natural numbers is:

\(S = \frac{n(n+1)(2n+1)}{6}\)

The average of the squares of the first n natural numbers is:

\(\text{Average} = \frac{S}{n} = \frac{(n+1)(2n+1)}{6}\)

Calculations:

Here, n = 47.

Average = \(\frac{(47+1)(2 \times 47+1)}{6}\)

= \(\frac{48 \times 95}{6}\)

= \(\frac{4560}{6}\)

= 760

The average of the squares of the first 47 natural numbers is 760.

Given:

The average of the squares of the first 46 natural numbers

Formula used:

The formula to find the average of squares of the first n natural numbers is:

\( = \dfrac{(n(n+1)(2n+1))}{6n}\)

Calculations:

n = 46

Average of squares = \(\dfrac{46(46+1)(2\times46+1)}{6\times46}\)

⇒ \(\dfrac{46 \times 47 \times 93}{6 \times 46}\)

⇒ \(\dfrac{47 \times 93}{6}\)

⇒ \(\dfrac{4371}{6}\)

⇒ 728.5

∴ The correct answer is option 2.

Given:

First 45 natural numbers.

Formula Used:

Average of the squares of the first n natural numbers = (Sum of squares of the first n natural numbers) / n

Sum of squares of the first n natural numbers = n × (n + 1) × (2n + 1) / 6

Calculation:

n = 45

Sum of squares of the first 45 natural numbers:

(45 × (45 + 1) × (2 × 45 + 1)) / 6

(45 × 46 × 91) / 6

⇒ 188370 / 6

⇒ 31395

Average of the squares of the first 45 natural numbers: 31395 / 45

⇒ 697.67

The average of the squares of the first 45 natural numbers is 697.67.

Given:

First 45 natural numbers.

Formula Used:

Average of the squares of the first n natural numbers = (Sum of squares of the first n natural numbers) / n

Sum of squares of the first n natural numbers = n × (n + 1) × (2n + 1) / 6

Calculation:

n = 45

Sum of squares of the first 45 natural numbers:

(45 × (45 + 1) × (2 × 45 + 1)) / 6

(45 × 46 × 91) / 6

⇒ 188370 / 6

⇒ 31395

Average of the squares of the first 45 natural numbers: 31395 / 45

⇒ 697.67

The average of the squares of the first 45 natural numbers is 697.67.

Given:

The average of the squares of the first 46 natural numbers

Formula used:

The formula to find the average of squares of the first n natural numbers is:

\( = \dfrac{(n(n+1)(2n+1))}{6n}\)

Calculations:

n = 46

Average of squares = \(\dfrac{46(46+1)(2\times46+1)}{6\times46}\)

⇒ \(\dfrac{46 \times 47 \times 93}{6 \times 46}\)

⇒ \(\dfrac{47 \times 93}{6}\)

⇒ \(\dfrac{4371}{6}\)

⇒ 728.5

∴ The correct answer is option 2.

Given:

The first 47 natural numbers.

Formula used:

The sum of squares of the first n natural numbers is:

\(S = \frac{n(n+1)(2n+1)}{6}\)

The average of the squares of the first n natural numbers is:

\(\text{Average} = \frac{S}{n} = \frac{(n+1)(2n+1)}{6}\)

Calculations:

Here, n = 47.

Average = \(\frac{(47+1)(2 \times 47+1)}{6}\)

= \(\frac{48 \times 95}{6}\)

= \(\frac{4560}{6}\)

= 760

The average of the squares of the first 47 natural numbers is 760.

Given:

First 48 natural numbers.

Concept used:

Sum of squares of first n natural Number = [n (n + 1) (2n + 1)]/6

Average of sum of squares of first n natural Number = [(n + 1) (2n + 1)]/6

Calculation:

Sum of squares of first 48 natural Number = [(48 + 1)(2 × 48 + 1)]/6 = 792.17

∴ The average of squares of first 48 natural numbers is 792.17

Given:

First 125 natural numbers.

Formula Used:

Average of first n natural numbers = (n + 1) / 2

Calculation:

n = 125

Average = (125 + 1) / 2

⇒ Average = 126 / 2

⇒ Average = 63

The average of the first 125 natural numbers is 63.