Chapter 5: Arithmetic Progressions

Key Formulas:
• nth term: aₙ = a + (n-1)d
• Sum of n terms: Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l)
• Common difference: d = aₙ - aₘ⁺⁴

Exercise 5.1

Q2. Write first 4 terms of AP: (i) a=10,d=10 (ii) a=-2,d=0 (iii) a=4,d=-3 (iv) a=-1,d=1/2 (v) a=-1.25,d=-0.25

Solution

(i) 10, 20, 30, 40
(ii) -2, -2, -2, -2
(iii) 4, 1, -2, -5
(iv) -1, -1/2, 0, 1/2
(v) -1.25, -1.50, -1.75, -2.00

Q3. Write first term and common difference: (i) 3, 1, -1, -3... (ii) -5, -1, 3, 7... (iii) 1/3, 5/3, 9/3... (iv) 0.6, 1.7, 2.8...

Solution

(i) a=3, d=-2
(ii) a=-5, d=4
(iii) a=1/3, d=4/3
(iv) a=0.6, d=1.1

Q4. Which are APs? (i) 2,4,8,16... (ii) 2,5/2,3,7/2... (iii) -1.2,-3.2,-5.2,-7.2... (iv) -10,-6,-2,2...

Solution

(i) Not AP (d not constant: 2,4,8)
(ii) AP, d=1/2, next 3 terms: 4, 9/2, 5
(iii) AP, d=-2, next 3 terms: -9.2, -11.2, -13.2
(iv) AP, d=4, next 3 terms: 6, 10, 14

Exercise 5.2

Q2. 30th term of AP: 10, 7, 4... is: (A) 97 (B) 77 (C) -77 (D) -87

Solution

a=10, d=-3
aₙ⁴₄=10+29(-3)=10-87=-77
∴ (C) -77

Q4. Which term of AP: 3, 8, 13, 18... is 78?

Solution

78=3+(n-1)5
75=5(n-1) → n-1=15 → n=16
∴ 16th term

Q5. Find number of terms: 7, 13, 19, ..., 205

Solution

205=7+(n-1)6
198=6(n-1) → n-1=33 → n=34
∴ 34 terms

Q8. Find the 20th term if 5th term is 21 and d=6.

Solution

a+4(6)=21 → a=21-24=-3
aₙ⁴⁵=-3+19(6)=-3+114=111

Q14. How many three-digit numbers are divisible by 7?

Solution

AP: 105, 112, ..., 994
994=105+(n-1)7
889=7(n-1) → n-1=127 → n=128
∴ 128 numbers

Q16. Sum of first 22 terms with d=7 and aₙ⁴⁵=149.

Solution

a+21(7)=149 → a=2
Sₙ⁴⁵=22/2[2(2)+21(7)]=11×151=1661

Q18. Rs 700 given as 7 prizes, each Rs 20 less than preceding. Find each prize.

Solution

Sₙ=7/2[2a+6(-20)]=700
7/2(2a-120)=700
7(a-60)=700 → a=160
∴ Prizes: Rs 160, 140, 120, 100, 80, 60, 40

Exercise 5.3

Q1. Find sum: 2, 7, 12, ..., to 10 terms

Solution

a=2, d=5, n=10
Sₙ=10/2[2(2)+9(5)]=5[4+45]=5×49=245

Q4. How many terms of AP: 9, 17, 25... give sum 636?

Solution

636=n/2[18+(n-1)8]
1272=n(10n+8)
8n²+10n-1272=0
4n²+5n-636=0
n=(-5+√(25+10176))/8=(-5+101)/8=12
∴ n=12

Q5. First term 5, last term 45, sum 400. Find n and d.

Solution

400=n/2(5+45)=25n → n=16
45=5+15d → d=40/15=8/3

Q7. Sum of first 40 positive integers divisible by 6.

Solution

AP: 6, 12, 18, ..., 240
Sₙ=40/2[2(6)+39(6)]=20[12+234]=20×246=4920

Q9. 600 units in 3rd year, 700 in 7th year. Find (i) 1st year (ii) 10th year (iii) total in 7 years.

Solution

a+2d=600, a+6d=700
4d=100 → d=25, a=550
(i) 550 units
(ii) 550+9(25)=775 units
(iii) Sₙ=7/2(550+700)=7/2×1250=4375 units

Q10. Sum of first 24 terms of AP: aₙ=3+4n.

Solution

aₙ=3+4n
a=3+4=7, d=4
Sₙ=24/2[2(7)+23(4)]=12[14+92]=12×106=1272

Exercise 5.4 (Optional)

Q5. Students plant trees: class k plants k per section, 3 sections per class, classes 1-12. Total trees?

Solution

Total = 3×(1+2+3+...+12)
= 3×12/2×(1+12)
= 3×6×13
= 234 trees

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