ARITHMETIC GEOMETRIC AND HARMONIC PROGRESSION
Arithmetic Geometric and Harmonic Progression :
In this section, we will learn, how to check the nth term of the given sequence is arithmetic progression, geometric progression or harmonic progression.
Arithmetic Geometric and Harmonic Progression - Example
Example 1 :
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 tn = 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,............
Since the common ratio is same, it is GP.
(ii) (n + 1)(n + 2) / (n + 3)(n + 4)
Solution :
Let tn = (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 tn = 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, .................
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.
(v) (2n+3) / (3n+4)
Solution :
Let tn = (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 tn = (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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