ARITHMETIC GEOMETRIC AND HARMONIC PROGRESSION

About "Arithmetic Geometric and Harmonic Progression"

Arithmetic Geometric and Harmonic Progression :

Here we are going to see how to check the nth term of the given sequence is arithmetic progression, geometric progression or harmonic progression.

Arithmetic Geometric and Harmonic Progression - Practice questions

Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them.

(i) 1/2ⁿ⁺¹

(ii) (n + 1)(n + 2) / (n + 3)(n + 4)

(iii) 4 (1/2)ⁿ

(iv) (−1)ⁿ/n

(v) (2n+3) / (3n+4)

(vi) 2018

(vii) (3n−2)/(3ⁿ⁻¹)

Solution :

(i) 1/2ⁿ⁺¹

Let t_n = 1/2ⁿ⁺¹

n = 1

t₁ = 1/2¹⁺¹

= 1/4

n = 2

t₂ = 1/2²⁺¹

= 1/8

n = 3

t₃ = 1/2³⁺¹

= 1/16

n = 4

t₄ = 1/2⁴⁺¹

= 1/32

n = 5

t₅ = 1/2⁵⁺¹

= 1/64

n = 6

t₆ = 1/2⁶⁺¹

= 1/128

First 6 terms of the sequence are,

1/4, 1/8, 1/16, 1/32, 1/64, 1/128,............

(1/2)², (1/2)³, (1/2)⁴, (1/2)⁵, (1/2)⁶, (1/2)⁷,..........

Hence the given n^th term will form a harmonic progression.

(ii) (n + 1)(n + 2) / (n + 3)(n + 4)

Solution :

Let t_n = (n + 1)(n + 2) / (n + 3)(n + 4)

n = 1

t₁ = (1+1)(1+2)/(1+3)(1+4)

= 6/20

n = 2

t₁ = (2+1)(2+2)/(2+3)(2+4)

= 12/30

n = 3

t₃ = (3+1)(3+2)/(3+3)(3+4)

= 20/42

n = 4

t₄ = (4+1)(4+2)/(4+3)(4+4)

= 30/56

n = 5

t₃ = (5+1)(5+2)/(5+3)(5+4)

= 42/72

n = 6

t₃ = (6+1)(6+2)/(6+3)(6+4)

= 56/90

First 6 terms of the sequence are,

6/20, 12/30, 20/42, 30/56, 42/72, 56/90,............

Common difference is not same, so it is not A.P

Common ratio is not same, so it is not G.P

It is not H.P

Hence the answer is none of them.

(iii) 4 (1/2)ⁿ

Solution :

Let t_n = 4 (1/2)ⁿ

n = 1

t₁ = 4 (1/2)ⁿ

= 4(1/2)¹

= 2

n = 2

t₂ = 4 (1/2)ⁿ

= 4(1/2)²

= 1

n = 3

t₃ = 4 (1/2)ⁿ

= 4(1/2)³

= 1/2

n = 4

t₄ = 4 (1/2)⁴

= 1/4

n = 5

t₅ = 4 (1/2)⁵

= 1/8

n = 6

t₆ = 4 (1/2)⁶

= 1/16

First 6 terms of the sequence are,

2, 1, 1/2, 1/4, 1/8, 1/16,....................

The common ratio is same, so it is G.P

(iv) (−1)ⁿ/n

Solution :

n = 1

t₁ = -1/1

= -1

n = 2

t₂ = 1/2

n = 3

t₂ = -1/3

n = 4

t₄ = 1/4

n = 5

t₅ = -1/5

n = 6

t₆ = 1/6

First 6 terms of the sequence are,

-1, 1/2, -1/3, 1/4, -1/5, 1/6, .................

The common ratio is same, so it is G.P

(v) (2n+3) / (3n+4)

Solution :

Let t_n = (2n+3) / (3n+4)

n = 1

t₁= 5/7

n = 2

t₂= 7/10

n = 3

t₃= 9/13

n = 4

t₄= 11/16

n = 5

t₅ = 13/19

n = 6

t₆ = 15/22

First 6 terms of the sequence are,

5/7, 7/10, 9/13, 11/16, 13/19, 15/22,....................

Hence the answer is none of these.

(vi) 2018

Solution :

The answer is none of these.

(vii) (3n−2)/(3ⁿ⁻¹)

Solution :

Let t_n = (3n−2)/(3ⁿ⁻¹)

n = 1

t₁= 1/1= 1

n = 2

t₂= 4/3

n = 3

t₃= 7/9

n = 4

t₄= 10/27

n = 5

t₅ = 13/81

n = 6

t₆ = 16/243

First 6 terms of the sequence are,

1, 4/3, 7/9, 10/27, 13/81, 16/243, .............

Hence the answer is arithmetico-geometric progression.

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