Question

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The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. [Hint : Sx-1 = S49 – Sx ]​

Answer 1

$\huge \red {\boxed {\boxed {\mathbb {QUESTION}}}}$

The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. [Hint : Sx-1 = S49 – Sx ]

$\huge \red {\boxed {\boxed {\mathbb {ANSWER}}}}$

Let there be a value of x such that the sum of the number of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it

House H

1

,H

2

,H

3

.......H

x−1

,H

x+1

........H

49

House No. 1 2 3 .......... x -1 x + 1..............49

House number will form an A.P whose first term and the common difference is 1

Sum of n terms S=

2

n

[2a+(n−1)×d]

S

x−1

=S

49

−S

x

⇒

2

(x−1)

[2(1)+(x−1−1)1]=

2

49

[2+48]−

2

x

[2(1)+(x−1)1]

⇒

2

x−1

[2+(x−2)]=

2

49

[50]−

2

x

[2+x−1]

⇒

2

x−1

[x]=

2

49

[50]−

2

x

[x+1]

⇒

2

x

[x−1+x+1]=

2

49

[50]

⇒

2

x

[2x]=49×25

⇒x

2 =49×25

⇒x=7×5=35.

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Answer 2

Answer:

Let the length and breadth of the park be L and B.

Perimeter of the rectangular park = 2 (L + B) = 80

So, L + B = 40

Or, B = 40 – L

Area of the rectangular park = L × B = L(40 – L) = 40L – L2 = 400

L2 – 40 L + 400 = 0,

which is a quadratic equation.

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

a = 1, b = -40, c = 400

Since, Discriminant = b2 – 4ac

=>(-40)2 – 4 × 400

=> 1600 – 1600

= 0

Thus, b2 – 4ac = 0

Therefore, this equation has equal real roots. Hence, the situation is possible.

Root of the equation,

L = –b/2a

L = (40)/2(1) = 40/2 = 20

Therefore, length of rectangular park, L = 20 m

And breadth of the park, B = 40 – L = 40 – 20 = 20 m.