FINDING Nth TERM OF A GEOMETRIC SEQUENCE

Write down formulae for the n^th term of these sequences :

Problem 1 :

3, 6, 12, 24, …

Solution :

3, 6, 12, 24, …

Since the common ratio is same, the given sequence is geometric sequence.

n^th term :

a_n = a₁ ⋅ r^{n - 1}

a_n = 3 ⋅ 2^{n - 1}

n^th term of the sequence :

a_n = 3  ⋅ 2^{n - 1}

Problem 2 :

36, 18, 9, 4.5, …

Solution :

36, 18, 9, 4.5, …

Since the common ratio is same, the given sequence is geometric sequence.

a_n = a₁  ⋅ r^{n - 1}

a_n = 36 ⋅ 1/2^{n - 1}

n^th term of the sequence :

a_n = 36  ⋅ (1/2)^{n - 1}

Problem 3 :

2, -6, 18, -54, …

Solution :

2, -6, 18, -54, …

It is geometric progression.

a_n = a₁ ⋅ r^{n - 1}

a_n = 2 ⋅ (-3)^{n - 1}

n^th term of the sequence :

a_n = 2 ⋅ (-3)^{n - 1}

Problem 4 :

90, -30, 10, -3 1/3, …

Solution :

90, -30, 10, -3 1/3, …

It is geometric progression.

a_n = a₁ ⋅ r^{n - 1}

a_n = 90  ⋅ (-1/3)^{n - 1}

n^th term of the sequence :

a_n = 90  ⋅ (-1/3)^{n - 1}

Problem 5 :

10, 100, 1000, …

Solution :

10, 100, 1000, …

It is geometric progression.

a_n = a₁  ⋅ r^{n - 1}

a_n = 10  ⋅ 10^{n - 1}

n^th term of the sequence :

a_n = 10 ⋅ (10)^{n - 1}

Problem 6 :

6, -6, 6, -6, …

Solution :

6, -6, 6, -6, …

It is geometric progression.

a_n = a₁  ⋅ r^{n - 1}

a_n = 6  ⋅ (-1)^{n - 1}

n^th term of the sequence :

a_n = 6  ⋅ (-1)^{n - 1}

Problem 7 :

1/4, 1/12, 1/36, 1/108, …

Solution :

1/4, 1/12, 1/36, 1/108, …

It is geometric progression.

a_n = a₁  ⋅ r^{n - 1}

a_n = 1/4  ⋅ (1/3)^{n - 1}

n^th term of the sequence :

a_n = ((1/3)^{n – 1})/4