The houses in a row numbered consecutively from 1 to 49. Show that there exists a value of X such that sum of numbers of houses preceding the house numbered X is equal to the sum of the numbers of houses following X.

Solution

Let there be 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.
That is , 1+2+3+......+(x-1) =(x-1)+(x+2)+.....+49
therefore,
1+2+3+.....+(x-1)
=[1+2+.....+x+(x+1)+.....+49]-(1+2+3+....+x)
$therefore space fraction numerator straight x minus 1 over denominator 2 end fraction left square bracket 1 plus straight x minus 1 right square bracket space equals 49 over 2 left square bracket 1 plus 49 right square bracket minus straight x over 2 left square bracket 1 plus 2 right square bracket therefore straight x left parenthesis straight x minus 1 right parenthesis equals 49 space straight x 50 minus straight x left parenthesis 1 plus straight x right parenthesis therefore straight x left parenthesis straight x minus 1 right parenthesis plus 1 left parenthesis 1 plus straight x right parenthesis equals 49 straight x 50 straight x squared minus straight x plus straight x plus straight x squared space equals space 49 space straight x 50 therefore space straight x squared space equals 49 space straight x 25 solving space saqure straight x space equals space 7 space straight x 5 space equals 35$
Since x is not a fraction, the value of x satisfying the given condition exists and is equal to 35.

Some More Questions From Arithmetic Progressions Chapter

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Arithmetic Progressions

The houses in a row numbered consecutively from 1 to 49. Show that there exists a value of X such that sum of numbers of houses preceding the house numbered X is equal to the sum of the numbers of houses following X.

Some More Questions From Arithmetic Progressions Chapter

Write first four terms of the AP, when the first term a and the common difference d are given as follows:

a = – 1.25, d = – 0.25

For the following APs, write the first term and the common difference
3, 1, – 1, – 3, . . .

For the following APs, write the first term and the common difference
– 5, – 1, 3, 7, . . .

For the following APs, write the first term and the common difference
0.6, 1.7, 2.8, 3.9, . . .

Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
– 1.2, – 3.2, – 5.2, – 7.2, . . .

Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
–10, –6, –2, 2, ....

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Arithmetic Progressions

The houses in a row numbered consecutively from 1 to 49. Show that there exists a value of X such that sum of numbers of houses preceding the house numbered X is equal to the sum of the numbers of houses following X.

Some More Questions From Arithmetic Progressions Chapter

Write first four terms of the AP, when the first term a and the common difference d are given as follows:a = – 1.25, d = – 0.25

For the following APs, write the first term and the common difference3, 1, – 1, – 3, . . .

For the following APs, write the first term and the common difference– 5, – 1, 3, 7, . . .

For the following APs, write the first term and the common difference0.6, 1.7, 2.8, 3.9, . . .

Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.– 1.2, – 3.2, – 5.2, – 7.2, . . .

Which of the following are APs ? If they form an AP, find the common difference d and write three more terms. –10, –6, –2, 2, ....

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Write first four terms of the AP, when the first term a and the common difference d are given as follows:

a = – 1.25, d = – 0.25

For the following APs, write the first term and the common difference
3, 1, – 1, – 3, . . .

For the following APs, write the first term and the common difference
– 5, – 1, 3, 7, . . .

For the following APs, write the first term and the common difference
0.6, 1.7, 2.8, 3.9, . . .

Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
– 1.2, – 3.2, – 5.2, – 7.2, . . .

Which of the following are APs ? If they form an AP, find the common difference d and write three more terms.
–10, –6, –2, 2, ....