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Q1(ii):
In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
(ii) The amount of air present in a cylinder when a vacuum pump removes $\frac{1}{4}$ of the air remaining in the cylinder at a time.
Solution :
Given: A cylinder contains an initial amount of air. A vacuum pump removes $\frac{1}{4}$ of the air remaining in the cylinder at each stroke.
To Find: Determine whether the sequence of the amounts of air remaining in the cylinder after each stroke forms an Arithmetic Progression (AP).
Step 1: Defining the Variables
Let the initial amount of air present in the cylinder be $V$ units.
Let $a_1$ be the amount of air after the 0th stroke (initial state).
Let $a_2$ be the amount of air after the 1st stroke.
Let $a_3$ be the amount of air after the 2nd stroke.
Let $a_4$ be the amount of air after the 3rd stroke.
Step 2: Calculating the sequence of air amounts
The pump removes $\frac{1}{4}$ of the air present in the cylinder at each step.
Initial amount: $a_1 = V$
After the 1st stroke ($a_2$):
$a_2 = V - \frac{1}{4}V = \frac{3}{4}V$
After the 2nd stroke ($a_3$):
The pump removes $\frac{1}{4}$ of the air remaining, which is $a_2$.
$a_3 = a_2 - \frac{1}{4}a_2 = \frac{3}{4}a_2$
Substituting $a_2 = \frac{3}{4}V$:
$a_3 = \frac{3}{4} \times (\frac{3}{4}V) = \frac{9}{16}V$
After the 3rd stroke ($a_4$):
$a_4 = a_3 - \frac{1}{4}a_3 = \frac{3}{4}a_3$
Substituting $a_3 = \frac{9}{16}V$:
$a_4 = \frac{3}{4} \times (\frac{9}{16}V) = \frac{27}{64}V$
Step 3: Checking for Arithmetic Progression
A sequence is an Arithmetic Progression if the difference between consecutive terms is constant (i.e., $a_{n+1} - a_n = d$, where $d$ is the common difference).
Calculate the first difference ($d_1$):
$d_1 = a_2 - a_1 = \frac{3}{4}V - V = -\frac{1}{4}V$
Calculate the second difference ($d_2$):
$d_2 = a_3 - a_2 = \frac{9}{16}V - \frac{3}{4}V$
To subtract, find a common denominator (16):
$d_2 = \frac{9}{16}V - \frac{12}{16}V = -\frac{3}{16}V$
Step 4: Comparison and Conclusion
Since $d_1 \neq d_2$ (because $-\frac{1}{4}V \neq -\frac{3}{16}V$), the difference between consecutive terms is not constant.
[Definition of an Arithmetic Progression: A sequence of numbers is an AP if the difference between any two consecutive terms is constant.]
Final Answer: The list of numbers does not form an Arithmetic Progression because the difference between consecutive terms is not constant.
More Questions from Class 10 Mathematics Arithmetic Progression EXERCISE 5.1
- Q1(i): In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (i) The taxi fare after each km when the fare is ₹ 15 for the first km and ₹ 8 for each additional km.
- Q1(iii): In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (iii) The cost of digging a well after every metre of digging, when it costs ₹ 150 for the first metre and rises by ₹ 50 for each subsequent metre.
- Q1(iv): In which of the following situations, does the list of numbers involved make an arithmetic progression, and why? (iv) The amount of money in the account every year, when ₹ 10000 is deposited at compound interest at 8 % per annum.
- Q2(i): Write first four terms of the AP, when the first term $a$ and the common difference $d$ are given as follows: (i) $a = 10, d = 10$
- Q2(ii): Write first four terms of the AP, when the first term $a$ and the common difference $d$ are given as follows: (ii) $a = –2, d = 0$
- Q2(iii): Write first four terms of the AP, when the first term $a$ and the common difference $d$ are given as follows: (iii) $a = 4, d = – 3$
- Q2(iv): Write first four terms of the AP, when the first term $a$ and the common difference $d$ are given as follows: (iv) $a = – 1, d = \frac{1}{2}$
- Q2(v): Write first four terms of the AP, when the first term $a$ and the common difference $d$ are given as follows: (v) $a = – 1.25, d = – 0.25$
- Q3(i): For the following APs, write the first term and the common difference: (i) 3, 1, – 1, – 3, . . .
- Q3(ii): For the following APs, write the first term and the common difference: (ii) – 5, – 1, 3, 7, . . .
- Q3(iii): For the following APs, write the first term and the common difference: (iii) $\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, . . .$
- Q3(iv): For the following APs, write the first term and the common difference: (iv) 0.6, 1.7, 2.8, 3.9, . . .
- Q4(i): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (i) 2, 4, 8, 16, . . .
- Q4(ii): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (ii) $2, \frac{5}{2}, 3, \frac{7}{2}, . . .$
- Q4(iii): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (iii) – 1.2, – 3.2, – 5.2, – 7.2, . . .
- Q4(iv): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (iv) – 10, – 6, – 2, 2, . . .
- Q4(ix): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (ix) 1, 3, 9, 27, . . .
- Q4(v): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (v) $3, 3+\sqrt{2}, 3+2\sqrt{2}, 3+3\sqrt{2}, . . .$
- Q4(vi): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (vi) 0.2, 0.22, 0.222, 0.2222, . . .
- Q4(vii): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (vii) 0, – 4, – 8, –12, . . .
- Q4(viii): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (viii) $-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, . . .$
- Q4(x): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (x) $a, 2a, 3a, 4a, . . .$
- Q4(xi): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (xi) $a, a^2, a^3, a^4, . . .$
- Q4(xii): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (xii) $\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, . . .$
- Q4(xiii): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (xiii) $\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, . . .$
- Q4(xiv): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (xiv) $1^2, 3^2, 5^2, 7^2, . . .$
- Q4(xv): Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms. (xv) $1^2, 5^2, 7^2, 73, . . .$
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