Chapter 9: Arithmetic Progressions Exercise – 9.1

Question: 1

Write the first terms of each of the following sequences whose n^th term are

(i) a_n = 3n + 2

(iii) a_n = 3_n

(v) a_n = (-1)ⁿ2ⁿ

(vii) a_n = n² - n + 1

(viii) a_n = n² - n + 1

Solution:

We have to write first five terms of given sequences

(i) a_n = 3_n + 2

Given sequence a_n = 3_n + 2

To write first five terms of given sequence put n = 1, 2, 3, 4, 5, we get

a₁ = (3 × 1) + 2 = 3 + 2 = 5

a₂ = (3 × 2) + 2 = 6 + 2 = 8

a₃ = (3 × 3) + 2 = 9 + 2 = 11

a₄ = (3 × 4) + 2 = 12 + 2 = 14

a₅ = (3 × 5) + 2 = 15 + 2 = 17

∴ The required first five terms of given sequence a_n = 3n + 2 are 5, 8, 11, 14, 17.

Given sequence

Put n = 1, 2, 3, 4, 5 then we get

∴ The required first five terms of given sequence

(iii) a_n = 3ⁿ

Given sequence a_n = 3ⁿ

To write first five terms of given sequence, put n = 1, 2, 3, 4, 5 in given sequence then,

a₁ = 3¹ = 3; a₂ = 3² = 9; a₃ = 27; a₄ = 3⁴ = 81; a₅ = 3⁵ = 243.

Given sequence,

To write first five terms, put n = 1, 2, 3, 4, 5 in given sequence

Then, we get

∴ The required first five terms are 1/5, 4/5, 7/5, 10/5, 13/5

(v) a_n = (-1)ⁿ2ⁿ

Given sequence is a_n = (-1)ⁿ2ⁿ

To get first five terms of given sequence an, put n = 1, 2, 3, 4, 5.

a₁ = (-1)¹.2¹ = (-1).2 = -2

a₂ = (-1)².2² = (-1).4 = -4

a₃ = (-1)³.2³ = (-1).8 = -8

a₄ = (-1)⁴.2⁴ = (-1).16 = 16

a₅ = (-1)⁵.2⁵ = (-1).32 = -32

∴ The first five terms are – 2, 4, - 8, 16, - 32.

The given sequence is,

To write first five terms of given sequence

Put n = 1, 2, 3, 4, 5. Then, we get

∴  The required first five terms are

(vii) a_n = n² - n + 1

The given sequence is, a_n = n² - n + 1

To write first five terms of given sequence a_n1 we get put n = 1, 2, 3, 4, 5. Then we get

a₁ = 1² - 1 + 1 = 1

a₂ = 2² - 2 + 1 = 3

a₃ = 3² - 3 + 1 = 7

a₄ = 4² - 4 + 1 = 13

a₅ = 5² - 5 + 1 = 21

∴ The required first five terms of given sequence a_n = n² - n + 1 are 1, 3, 7, 13, 21

(viii) a_n = 2n² - 3n + 1

The given sequence is a_n = 2n² - 3n + 1

To write first five terms of given sequence and, we put n = 1, 2, 3, 4, 5. Then we get

a₁ = 2.1² - 3.1 + 1 = 2 - 3 + 1 = 0

a₂ = 2.2² - 3.2 + 1 = 8 - 6 + 1 = 3

a₃ = 2.3² - 3.3 + 1 = 18 - 9 + 1 = 10

a₄ = 2.4² - 3.4 + 1 = 32 - 12 + 1 = 21

a₅ = 2.5² - 3.5 + 1 = 50 - 15 + 1 = 36

∴ The required first five terms of given sequence a_n - 2n² - 3n + 1 are 0, 3, 10, 21, 36

Given sequence is,

To write first five terms of given sequence we put n = 1, 2, 3, 4, 5. Then, we get,

∴ The required first five terms of given sequence

Question: 2

Find the indicated terms in each of the following sequences whose nth terms are:

(i) a_n = 5n - 4; a₁₂ and a₁₅

(iii) a_n = n(n - 1)(n - 2); a₅ and a₈

(iv) a_n = (n - 1)(2 - n)(3 + n);a₁₁ a₂₁ a₃

(v) an = (-1)ⁿn;a₃, a₅, a₈

Solution:

We have to find the required term of a sequence when nth term of that sequence is given:

(i) a_n = 5n - 4; a₁₂ and a₁₅

Given n^th term of a sequence an = 5_n - 4

To find 12^th terms, 15^th terms of that sequence, we put n = 12 15 in its n^th term.

Then, we get

a₁₂ = 5.12 - 4 = 60 - 4 = 56

a₁₅ = 5.15 - 4 = 15 - 4 = 71

∴ The required terms a₁₂ = 56, a₁₅ = 71

Given nth term is

To find 7^th, 8^th terms of given sequence, we put n = 7, 8.

∴ The required terms

(iii) a_n = n(n - 1)(n - 2); a₅ and a₈

Given n^th term is a_n = n(n - 1)(n - 2)

To find 5^th, 8^th terms of given sequence, put n = 5, 8 in an then, we get

a₅ = 5(5 - 1).(5 - 2) = 5.4.3 = 60

a₈ = 8.(8 - 1).(8 - 2) = (8.7.6) = 336

∴ The required terms are a₅ = 60 and a₈ = 336

(iv) a_n = (n - 1)(2 - n)(3 + n);a₁₁ a₂₁ a₃

The given n^th term is a_n = (n + 1)(2 - n)(3 + n)

To find a₁, a₂, a₃ of given sequence put n = 1, 2, 3 is an

a₁ = (1 - 1)(2 - 1)(3 + 1) = 0.1.4 = 0

a₂ = (2 - 1)(2 - 2)(3 + 2) = 1.0.5 = 0

a₃ = (3 - 1)(2 - 3)(3 + 3) = 2.-1.6 = -12

∴ The required terms a₁ = 0, a₂ = 0, a₃ = -12

(v) a_n = (-1)ⁿn; a₃, a₅, a₈

The given n^th term is, a_n = (-1)ⁿ.n

To find a₃, a₅, a₈ of given sequence put n = 3, 5, 8, in a_n.

a₃ = (-1)³.3 = -1.3 = -3

a₅ = (-1)⁵.5 = -1.5 = -5

a₈ = (-1)⁸ = 1.8 = 8

∴ The required terms a₃ = -3, a₅ = -5, a₈ = 8

Question: 3

Find the next five terms of each of the following sequence given by:

(i) a₁ = 1, 1_n = a_n-1 + 2, n ≥ 2

(ii) a₁ = a₂ = 2, a_n = a_n-1 - 3, n > 2

(iv) a₁ = 4, a_n = 4 a_n-1 + 3, n > 1

Solution:

We have to find next five terms of following sequences.

(i) a₁ = 1, a_n = a_n-1 + 2, n ≥ 3

Given first term (a₁) = 1,

n^th term a_n = a_n-1 + 2, n ≥ 2

To find 2^nd, 3^rd, 4^th, 5^th, 6^th terms, we use given condition n ≥ 2 for nth term a_n = a_n-1 + 2

a₂ = a_2-1 + 2 = a₁ + 2 = 1 + 2 = 3 (∴ a₁ = 1)

a₃ = a_3-1 + 2 = a₂ + 2 = 3 + 2 = 5

a₄ = a_4-1 + 2 = a₃ + 2 = 5 + 2 = 7

a₅ = a_5-1 + 2 = a₄ + 2 = 7 + 2 = 9

a₆ = a_6-1 + 2 = a₅ + 2 = a + 2 = 11

∴ The next five terms are,

a₂ = 3, a₃ = 5, a₄ = 7, a₅ = a, a₆ = 11

(ii) a₁ = a₂ = 2, a_n = a_n-1 - 3, n > 2

Given,

First term (a₁) = 2

Second term (a₂) = 2

n^th term (a_n) = a_n-1 - 3

To find next five terms i.e., a₃, a₄, a₅, a₆, a₇ we put n = 3, 4, 5, 6, 7 is a_n

a₃ = a_3-1 - 3 = 2 - 3 = – 1

a₄ = a_4-1 - 3 = a₃ - 3 = – 1 - 3 = – 4

a₅ = a_5-1 - 3 = a₄ - 3 = -4 - 3 = -7

a₆ = a_6-1 - 3 = a₅ - 3 = -7 - 3 = -10

a₇ = a_7-1 - 3 = a₆ - 3 = -10 - 3 = -13

∴ The next five terms are, a₃ = -1, a₄ = – 4, a₅ = -7, a₆ = -10, a₇ = -13

Given, first term (a₁) = – 1

To find next five terms i.e., a₂, a₃, a₄, a₅, a₆ we put n = 2, 3, 4, 5, 6 is a_n

∴ The next five terms are,

(iv) a1 = 4, an = 4 an-1 + 3, n > 1

Given,

First term (a₁) = 4

nth term (a_n) = 4 a_n-1 + 3, n > 1

To find next five terms i.e., a₂, a₃, a₄, a₅, a₆ we put n = 2, 3, 4, 5, 6 is a_n

Then, we get

a₂ = 4a_2-1 + 3 = 4. a₁ + 3 = 4.4 + 3 = 19 (∴ a₁ = 4)

a₃ = 4a_3-1 + 3 = 4.a₂ + 3 = 4(19) + 3 = 79

a₄ = 4a_4-1 + 3 = 4.a₃ + 3 = 4(79) + 3 = 319

a₅ = 4a_5-1 + 3 = 4.a₄ + 3 = 4(319) + 3 = 1279

a₆ = 4a_6-1 + 3 = 4.a₅ + 3 = 4(1279) + 3 = 5119

∴ The required next five terms are,

a₂ = 19, a₃ = 79, a₄ = 319, a₅ = 1279, a₆ = 5119