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प्रश्न
In the following situation, involved make an arithmetic progression? and why?
The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time.
In the following situation, the sequence of number formed will form an A.P.?
The amount of air present in the cylinder when a vacuum pump removes each time 1/4 of their remaining in the cylinder.
उत्तर १
Let the quantity of air in the cylinder be 1.
T1 = 1
Air removed = `1/4`
`"T"_2 = 1 - 1/4 = (4 - 1)/4 = 3/4`
Air removed = `3/4 xx 1/4 = 3/16`
`"T"_3 = 3/4 - 3/16 = (12 - 3)/16 = 9/16`
Air removed = `9/16 xx 1/4 = 9/64`
`"T"_4 = 9/16 - 9/64 = (36 - 9)/64 = 27/64`
Series: `1, 3/4, 9/16, 27/64`
`"d"_1 = 3/4 - 1 = (-1)/4`
`"d"_2 = 9/16 - 3/4 = (-3)/16`
Here the common difference is not the same hence it is A.P. Not there.
उत्तर २
Here, let us take the initial amount of air present in the cylinder as 100 units.
So,
Amount left after vacuum pump removes air for 1st time = `100 - (1/4) 100`
= 100 - 25
= 75
Amount left after vacuum pump removes air for 2nd time = `75 - (1/4)75`
= 75 - 18.75
= 56.25
Amount left after vacuum pump removes air for 3rd time = `56.25 - (1/4) 56.25`
= 56.25 - 14.06
= 42.19
Thus, the amount left in the cylinder at various stages is 100, 75, 56, 25, 42, 19
Now, for a sequence to be an A.P., the difference between adjacent terms should be equal.
Here
`a_1 - a = 75 - 100`
= -25
Also
`a_2 - a_1 = 56.25 - 75`
= -18.75
Since `a_1- a != a_2 - a_1`
The sequence is not an A.P.
संबंधित प्रश्न
State whether the following sequence is an AP or not: 1, 3, 6, 10………
Write the first three terms in each of the sequences defined by the following
an = 3n + 2
Four numbers are in A.P. If their sum is 20 and the sum of their square is 120, then find the middle terms
For the following arithmetic progressions write the first term a and the common difference d:
−1.1, −3.1, −5.1, −7.1, ...
Find the 10th term of the AP 1,4, 7, 10….
Is −150 a term of the A.P. 11, 8, 5, 2, ...?
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.
Find the middle term of the A.P. 213, 205, 197, ...., 37.
If the 5th term of an A.P. is 31 and 25th term is 140 more than the 5th term, find the A.P.
Find the first term and common difference of the Arithmetic Progressions whose nth term is given below
tn = 4 – 7n
Find d if t9 = 23 व a = 7
Which term of following A.P. is −940.
50, 40, 30, 20 ........
Activity :- Here a = `square`, d = `square`, tn = −940
According to formula, tn = a + (n − 1)d
−940 = `square`
n = `square`
Find first four terms of an A.P., whose first term is 3 and common difference is 4.
The nth term of an A.P. is given by an = 3 + 4n. The common difference is ______.
If (p + q)th term of an A.P. is m and (p - q)th term is n, then pth term is ______.
Which of the following form an AP? Justify your answer.
11, 22, 33,...
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Case Study Push-ups are a fast and effective exercise for building strength. These are helpful in almost all sports including athletics. While the push-up primarily targets the muscles of the chest, arms, and shoulders, support required from other muscles helps in toning up the whole body.
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- Form an A.P representing the number of push-ups per day and hence find the minimum number of days he needs to practice before the day his goal is accomplished?
- Find the total number of push-ups performed by Nitesh up to the day his goal is achieved.
The next term of the A.P. : `sqrt(6), sqrt(24), sqrt(54)` is ______.
Statement A (Assertion): `-5, (-5)/2, 0, 5/2`, .... is in Arithmetic Progression.
Statement R (Reason): The terms of an Arithmetic Progression cannot have both positive and negative rational numbers.
