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Q4(v):
Which of the following are APs ? If they form an AP, find the common difference $d$ and write three more terms.
(v) $3, 3+\sqrt{2}, 3+2\sqrt{2}, 3+3\sqrt{2}, . . .$
Solution :
Given: A sequence of numbers: $3, 3+\sqrt{2}, 3+2\sqrt{2}, 3+3\sqrt{2}, \dots$
To find: Determine if the sequence forms an Arithmetic Progression (AP). If it does, find the common difference $d$ and write the next three terms.
Step 1: Definition of an Arithmetic Progression
An Arithmetic Progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference ($d$).
Let the terms be $a_1, a_2, a_3, a_4, \dots$
Here, $a_1 = 3$, $a_2 = 3+\sqrt{2}$, $a_3 = 3+2\sqrt{2}$, $a_4 = 3+3\sqrt{2}$.
Step 2: Calculating the differences between consecutive terms
We calculate the difference $d_n = a_{n+1} - a_n$ for consecutive terms:
Difference 1 ($d_1$):
$d_1 = a_2 - a_1 = (3 + \sqrt{2}) - 3$
$d_1 = 3 - 3 + \sqrt{2} = \sqrt{2}$
Difference 2 ($d_2$):
$d_2 = a_3 - a_2 = (3 + 2\sqrt{2}) - (3 + \sqrt{2})$
$d_2 = 3 + 2\sqrt{2} - 3 - \sqrt{2}$
$d_2 = (3 - 3) + (2\sqrt{2} - \sqrt{2}) = \sqrt{2}$
Difference 3 ($d_3$):
$d_3 = a_4 - a_3 = (3 + 3\sqrt{2}) - (3 + 2\sqrt{2})$
$d_3 = 3 + 3\sqrt{2} - 3 - 2\sqrt{2}$
$d_3 = (3 - 3) + (3\sqrt{2} - 2\sqrt{2}) = \sqrt{2}$
Step 3: Verification
Since $d_1 = d_2 = d_3 = \sqrt{2}$, the difference between consecutive terms is constant. Therefore, the given sequence is an Arithmetic Progression with common difference $d = \sqrt{2}$.
Step 4: Finding the next three terms
To find the next terms, we add the common difference $d = \sqrt{2}$ to the last known term ($a_4 = 3 + 3\sqrt{2}$):
Fifth term ($a_5$):
$a_5 = a_4 + d = (3 + 3\sqrt{2}) + \sqrt{2} = 3 + 4\sqrt{2}$
Sixth term ($a_6$):
$a_6 = a_5 + d = (3 + 4\sqrt{2}) + \sqrt{2} = 3 + 5\sqrt{2}$
Seventh term ($a_7$):
$a_7 = a_6 + d = (3 + 5\sqrt{2}) + \sqrt{2} = 3 + 6\sqrt{2}$
Final Answer: The sequence forms an AP with common difference $d = \sqrt{2}$. The next three terms are $3+4\sqrt{2}, 3+5\sqrt{2},$ and $3+6\sqrt{2}$.
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(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.
- 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(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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