Question

In the sequence of 3 consecutive natural number, product of two alternate number is one less than the square of middle number.
1. 99×101=
2.49×51=
3. 199×201=​

Answer 1

Answer:

Step-by-step explanation:

Let the four consecutive natural numbers be x,x+1,x+2,x+3.

According to given condition,

x(x+1)(x+2)(x+3)=5040

x(x+3)(x+1)(x+2)=5040

(x

2

+3x)(x

2

+3x+2)=5040

Let x

2

+3x=a

a(a+2)=5040

a

2

+2a=5040

Adding 1 on both sides,

a

2

+2a+1=5040+1

(a+1)

2

=5041

Taking square roots on both sides,

a+1=±71

x

2

+3x−70=0 or x

2

+3x+72=0

The discriminant of the 2

nd

equation is negative, hence no real roots exist for that equation. Hence, we only solve the 1

st

equation.

x(x+10)−7(x+10)=0

(x+10)(x−7)=0

x+10=0 or x−7=0

x=−10 or x=7

−10 is not a natural number

∴x=−10 is not applicable

The numbers are 7,7+1=8,7+2=9,7+3=10

∴ The numbers are 7,8,9,10.