EXERCISE 5.2
Fill in the blanks using the formula:
Formula: aₙ = a + (n − 1)d
(i)
Given: a = 7, d = 3, n = 8
Step 1: Use formula
aₙ = a + (n − 1)d
Step 2: Substitute values
a₈ = 7 + (8 − 1) × 3
Step 3: Simplify
a₈ = 7 + (7 × 3)
a₈ = 7 + 21
a₈ = 28
(ii)
Given: a = −18, n = 10, aₙ = 0
Step 1: Use formula
0 = −18 + (10 − 1)d
Step 2: Simplify
0 = −18 + 9d
Step 3: Solve for d
18 = 9d
d = 2
(iii)
Given: d = −3, n = 18, aₙ = −5
Step 1: Use formula
−5 = a + (18 − 1)(−3)
Step 2: Simplify
−5 = a + 17(−3)
−5 = a − 51
Step 3: Solve for a
a = −5 + 51
a = 46
(iv)
Given: a = −18.9, d = 2.5, aₙ = 3.6
Step 1: Use formula
3.6 = −18.9 + (n − 1) × 2.5
Step 2: Simplify
3.6 + 18.9 = (n − 1) × 2.5
22.5 = (n − 1) × 2.5
Step 3: Solve
22.5 ÷ 2.5 = n − 1
9 = n − 1
n = 10
(v)
Given: a = 3.5, d = 0, n = 105
Step 1: Use formula
aₙ = 3.5 + (105 − 1) × 0
Step 2: Simplify
aₙ = 3.5 + 0
aₙ = 3.5
Final Answers:
(i) a₈ = 28
(ii) d = 2
(iii) a = 46
(iv) n = 10
(v) a₁₀₅ = 3.5
Exercise 5.2 – Stepwise Solutions
Formula Used: aₙ = a + (n − 1)d
2. Choose the correct option
(i) AP: 10, 7, 4, …
a = 10, d = −3
a₃₀ = 10 + (30 − 1)(−3)
= 10 + 29(−3)
= 10 − 87
= −77
Answer: (C) −77
(ii) AP: −3, −1/2, 2, …
d = (−1/2 + 3) = 5/2
a₁₁ = −3 + (11 − 1)(5/2)
= −3 + 10 × 5/2
= −3 + 25
= 22
Answer: (B) 22
3. Missing terms
(i) 2, __, 26
Let common difference = d
26 = 2 + 2d
24 = 2d
d = 12
Missing term = 2 + 12 = 14
(ii) __, 13, __, 3
Let first term = a
3 = a + 3d
13 = a + d
Subtract:
3 − 13 = (a + 3d) − (a + d)
−10 = 2d
d = −5
a = 13 − (−5) = 18
Missing terms: 18, 8
(iii) 5, __, __, 9 1/2
9.5 = 5 + 3d
4.5 = 3d
d = 1.5
Terms: 6.5, 8
(iv) −4, __, __, __, __, 6
6 = −4 + 5d
10 = 5d
d = 2
Terms: −2, 0, 2, 4
(v) __, 38, __, __, __, −22
Let first term = a
−22 = a + 5d
38 = a + d
Subtract:
−60 = 4d
d = −15
a = 38 − (−15) = 53
Terms: 53, 38, 23, 8, −7, −22
4.
AP: 3, 8, 13, 18, …
a = 3, d = 5
78 = 3 + (n − 1)5
75 = 5(n − 1)
n − 1 = 15
n = 16
5.
(i) 7, 13, 19, …, 205
a = 7, d = 6
205 = 7 + (n − 1)6
198 = 6(n − 1)
n − 1 = 33
n = 34
(ii) 18, 15 1/2, 13, …, −47
d = −2.5
−47 = 18 + (n − 1)(−2.5)
−65 = −2.5(n − 1)
n − 1 = 26
n = 27
6.
AP: 11, 8, 5, 2…
a = 11, d = −3
150 = 11 + (n − 1)(−3)
139 = −3(n − 1)
No natural solution
150 is not a term.
7.
a₁₁ = 38 → a + 10d = 38
a₁₆ = 73 → a + 15d = 73
Subtract:
5d = 35
d = 7
a = −32
a₃₁ = −32 + 30×7 = 178
8.
Given: a₃ = 12 → a + 2d = 12
a₅₀ = 106 → a + 49d = 106
Subtract:
47d = 94
d = 2
a = 8
a₂₉ = 8 + 28×2 = 64
9.
a₃ = 4 → a + 2d = 4
a₉ = −8 → a + 8d = −8
Subtract:
6d = −12
d = −2
a = 8
0 = 8 + (n − 1)(−2)
n = 5
10.
a₁₇ − a₁₀ = 7
[a +16d] − [a +9d] = 7
7d = 7
d = 1
11.
AP: 3, 15, 27, 39…
a = 3, d = 12
aₙ − a₅₄ = 132
(n − 54)12 = 132
n − 54 = 11
n = 65
12.
Difference between 100th terms = 100
Difference between 1000th terms = 100
13.
Smallest 3-digit multiple of 7 = 105
Largest = 994
n = 128
14.
Multiples of 4 between 10 and 250:
12 to 248
Total terms = 60
15.
63 + (n − 1)2 = 3 + (n − 1)7
60 = 5(n − 1)
n = 13
16.
a₃ = 16 → a + 2d = 16
a₇ − a₅ = 12
(a +6d) − (a +4d) = 12
2d = 12
d = 6
a = 4
Required AP: 4, 10, 16, 22, 28, …
Exercise 5.2 – Stepwise Solutions (Q17–20)
Formula Used:
aₙ = a + (n − 1)d
17. Find the 20th term from the last term of the AP: 3, 8, 13, …, 253
Step 1: Identify values
a = 3, d = 5, last term = 253
Step 2: Find total number of terms
253 = 3 + (n − 1)5
250 = 5(n − 1)
n − 1 = 50
n = 51
Total terms = 51
Step 3: 20th term from last = (51 − 20 + 1)th term
= 32nd term
Step 4: Find 32nd term
a₃₂ = 3 + (32 − 1)5
= 3 + 31×5
= 3 + 155
= 158
Answer: 158
18. Sum of 4th and 8th terms is 24 and sum of 6th and 10th terms is 44
Let first term = a, common difference = d
a₄ = a + 3d
a₈ = a + 7d
Equation (1):
(a + 3d) + (a + 7d) = 24
2a + 10d = 24
a + 5d = 12
a₆ = a + 5d
a₁₀ = a + 9d
Equation (2):
(a + 5d) + (a + 9d) = 44
2a + 14d = 44
a + 7d = 22
Step 2: Subtract equations
(a + 7d) − (a + 5d) = 22 − 12
2d = 10
d = 5
Step 3: Substitute d = 5
a + 5(5) = 12
a + 25 = 12
a = −13
First three terms:
−13, −8, −3
19. Subba Rao Salary Problem
Given:
a = 5000
d = 200
Target salary = 7000
Step 1:
7000 = 5000 + (n − 1)200
2000 = 200(n − 1)
n − 1 = 10
n = 11
Step 2: Year calculation
Starting year = 1995
11th year = 1995 + 10
= 2005
Answer: 2005
20. Ramkali Weekly Savings
Given:
a = 5
d = 1.75
aₙ = 20.75
Step 1:
20.75 = 5 + (n − 1)1.75
Step 2:
15.75 = 1.75(n − 1)
n − 1 = 9
n = 10
Answer: n = 10