Geometric Sequences - Mr-Mathematics.com - Free Printable

I will solve this problem by finding the common ratio (r) and the nth term for each of the four geometric sequences provided in the image.

The formula for the nth term of a geometric sequence is given as:
G.P. = ar^(n-1)
where 'a' is the first term and 'r' is the common ratio.

Let me analyze each sequence one by one.

Sequence a)

Sequence: 2, 4, 8, 16, 32, 64

Step 1: Find the common ratio (r)
To find the common ratio, I divide any term by the previous term.
r = 4 / 2 = 2
r = 8 / 4 = 2
r = 16 / 8 = 2
The common ratio r = 2.

Step 2: Find the nth term
Using the formula G.P. = ar^(n-1), with a = 2 and r = 2:
nth term = 2 * 2^(n-1) = 2^1 * 2^(n-1) = 2^(n)

So, the nth term is 2^n.

Sequence b)

Sequence: 3, 9, 81, 243, 729, 2187

Step 1: Find the common ratio (r)
r = 9 / 3 = 3
r = 81 / 9 = 9
Wait, this doesn't seem right. Let me check again.
Actually, looking at the sequence, it appears there might be an error in the image or my reading. Let me verify the pattern:
3 to 9 is *3
9 to 81 is *9
81 to 243 is *3
243 to 729 is *3
729 to 2187 is *3
This is not a consistent geometric sequence because the ratio changes from 3 to 9 between the second and third terms.

However, if we assume the third term should be 27 (which would make it 3, 9, 27, 81, 243, 729), then the common ratio would be 3. But since the image shows 81 as the third term, I will calculate based on what's written.

Actually, let me recalculate carefully:
From term 1 to term 2: 9/3 = 3
From term 2 to term 3: 81/9 = 9
From term 3 to term 4: 243/81 = 3
From term 4 to term 5: 729/243 = 3
From term 5 to term 6: 2187/729 = 3

This is not a standard geometric sequence as the ratio is not constant. There seems to be an inconsistency in the sequence provided. However, if we consider the sequence starting from the second term, it becomes geometric with ratio 3: 9, 27, 81, 243, 729, 2187 (but the third term is listed as 81, not 27).

Given the context of the problem, it's likely that the third term should be 27, making the sequence 3, 9, 27, 81, 243, 729, 2187 with a common ratio of 3.

But since the image clearly shows "81" as the third term, I will proceed with the calculation as written, noting the inconsistency.

If we force it to be geometric, the ratio is not constant. Let's assume it's a typo and the sequence is meant to be 3, 9, 27, 81, 243, 729, which is a geometric sequence with r=3.

For the sake of solving the problem as intended, I'll assume the sequence is 3, 9, 27, 81, 243, 729, 2187 with r=3.

Step 2: Find the nth term
With a = 3 and r = 3:
nth term = 3 * 3^(n-1) = 3^n

Sequence c)

Sequence: 16, 8, 4, 2, 1, 0.5

Step 1: Find the common ratio (r)
r = 8 / 16 = 0.5
r = 4 / 8 = 0.5
r = 2 / 4 = 0.5
r = 1 / 2 = 0.5
r = 0.5 / 1 = 0.5
The common ratio r = 0.5 or 1/2.

Step 2: Find the nth term
With a = 16 and r = 1/2:
nth term = 16 * (1/2)^(n-1)

Sequence d)

Sequence: 12, 6, 3, 1.5, 0.75, 0.375

Step 1: Find the common ratio (r)
r = 6 / 12 = 0.5
r = 3 / 6 = 0.5
r = 1.5 / 3 = 0.5
r = 0.75 / 1.5 = 0.5
r = 0.375 / 0.75 = 0.5
The common ratio r = 0.5 or 1/2.

Step 2: Find the nth term
With a = 12 and r = 1/2:
nth term = 12 * (1/2)^(n-1)

Summary of Solutions:

a) Common Ratio: 2, nth term: 2^n
b) Common Ratio: 3 (assuming the sequence is 3, 9, 27, 81, 243, 729), nth term: 3^n
c) Common Ratio: 0.5, nth term: 16 * (1/2)^(n-1)
d) Common Ratio: 0.5, nth term: 12 * (1/2)^(n-1)

Note: For sequence b, there appears to be an inconsistency in the provided sequence (the third term is 81 instead of the expected 27 for a geometric sequence with ratio 3). I have assumed it was meant to be 27 for the purpose of providing a valid geometric sequence solution.