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Q3(iii):
For the following APs, write the first term and the common difference:
(iii) $\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, . . .$
Solution :
Given: An Arithmetic Progression (AP) sequence: $\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, . . .$
To find: The first term ($a$) and the common difference ($d$) of the given AP.
Step 1: Identifying the First Term
By definition, an Arithmetic Progression is a sequence of numbers such that the difference between consecutive terms is constant. The first term is denoted by $a$ or $a_1$.
Looking at the sequence: $a_1 = \frac{1}{3}, a_2 = \frac{5}{3}, a_3 = \frac{9}{3}, a_4 = \frac{13}{3}$.
Therefore, the first term $a = \frac{1}{3}$.
Step 2: Defining the Common Difference
The common difference ($d$) of an AP is defined as the difference between any two consecutive terms. Mathematically, it is expressed as:
$d = a_{n} - a_{n-1}$ for $n > 1$.
Specifically, for this sequence:
$d = a_2 - a_1$
Step 3: Calculating the Common Difference
Substitute the values of $a_2$ and $a_1$ into the formula:
$d = \frac{5}{3} - \frac{1}{3}$
[Since the denominators are the same, we subtract the numerators directly]
$d = \frac{5 - 1}{3}$
$d = \frac{4}{3}$
Step 4: Verification (Optional but Recommended)
To ensure the sequence is indeed an AP, we verify the difference between the third and second terms:
$d = a_3 - a_2 = \frac{9}{3} - \frac{5}{3} = \frac{4}{3}$
[Since the difference is consistent, the value of $d$ is confirmed as $\frac{4}{3}$]
Final Answer: The first term $a = \frac{1}{3}$ and the common difference $d = \frac{4}{3}$.
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(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) $1^2, 5^2, 7^2, 73, . . .$
CBSE Solutions for Class 10 Mathematics Arithmetic Progression
Chapters in CBSE - Class 10 Mathematics
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