Question

1. The product of four consecutive natural numbers is 840. Find the numbers.​

Answer 1

In context to questions asked,

We have to determine the value of the numbers.

As per questions,

It is given that,

Product of four consecutive natural numbers = 840.

As we know that,

Counting numbers are called as Natural Numbers.

So, let the numbers be - X, (X+1), (X2) and (X+3)

So, from above

$x + (x + 1) + (x + 2) + (x + 3) = 840 \\ 4x + 6 = 840 \\ 4x = 840 - 6 \\ 4x = 834 \\ x = \frac{834}{4} = 208.5$

Hence, number will be 208.5, 209.5, 210.5 and 211.5 respectively.

Answer 2

Answer:

The four consecutive natural numbers are 4, 5, 6, 7.

Step-by-step-explanation:

Let the smallest of the four consecutive natural numbers be ( n - 1 ), ( n ≠ 1 ).

∴ The four consecutive natural numbers are ( n - 1 ), n, ( n + 1 ), ( n + 2 ).

From the given condition,

Product of four consecutive natural numbers is 840.

∴ ( n - 1 ) ( n ) ( n + 1 ) ( n + 2 ) = 840

⇒ ( n - 1 ) ( n + 1 ) * n ( n + 2 ) = 840

⇒ ( n² - 1² ) * ( n² + 2n ) = 840

⇒ ( n² - 1 ) * ( n² + 2n ) = 840

⇒ n² ( n² + 2n ) - 1 ( n² + 2n ) = 840

⇒ n⁴ + 2n³ - n² - 2n = 840

⇒ n⁴ + 2n³ - n² - 2n - 840 = 0

⇒ n⁴ - 5n³ + 7n³ - 35n² + 34n² - 170n + 168n - 840 = 0

⇒ n³ ( n - 5 ) + 7n² ( n - 5 ) + 34n ( n - 5 ) + 168 ( n - 5 ) = 0

⇒ ( n - 5 ) ( n³ + 7n² + 34n + 168 ) = 0

⇒ ( n - 5 ) ( n³ + 6n² + n² + 6n + 28n + 168 ) = 0

⇒ ( n - 5 ) [ n² ( n + 6 ) + n ( n + 6 ) + 28 ( n + 6 ) ] = 0

⇒ ( n - 5 ) [ ( n + 6 ) ( n² + n + 28 ) ] = 0

⇒ ( n - 5 ) ( n + 6 ) ( n² + n + 28 ) = 0

Now,

( n - 5 ) = 0

⇒ n - 5 = 0

⇒ n = 5

Now,

( n + 6 ) = 0

⇒ n + 6 = 0

⇒ n = - 6

As n is a natural number, n = - 6 is unacceptable.

Now,

n² + n + 28 = 0

Comparing with ax² + bx + c = 0, we get,

• a = 1
• b = 1
• c = 28

b² - 4ac = 1² - 4 * 1 * 28

⇒ b² - 4ac = 1 - 4 * 28

⇒ b² - 4ac = 1 - 112

⇒ b² - 4ac = - 111

b² - 4ac < 0

∴ n ∉ N

∴ The value of n is 5.

Now,

First number = ( n - 1 )

⇒ First number = 5 - 1

⇒ First number = 4

Now,

Second number = n

⇒ Second number = 5

Now,

Third number = ( n + 1 )

⇒ Third number = 5 + 1

⇒ Third number = 6

Now,

Fourth number = ( n + 2 )

⇒ Fourth number = 5 + 2

⇒ Fourth number = 7

∴ The four consecutive natural numbers are 4, 5, 6, 7.