The problem asks us to find the next three terms in each of the given sequences.

1. Problem Description

The problem asks us to find the next three terms in each of the given sequences.

2. Solution Steps

i) -8, 3, 14, 25, ...

The difference between consecutive terms are:

3 - (-8) = 11

14 - 3 = 11

25 - 14 = 11

Since the difference between consecutive terms is constant, this is an arithmetic sequence with a common difference of

1

1. Therefore, the next three terms are:

25 + 11 = 36

36 + 11 = 47

47 + 11 = 58

So, the next three terms are 36, 47, and

5

8.

ii) 1, 2, 8, 16, ...

This sequence can be interpreted in several ways. One possible pattern is:

1 = $2^0$

2 = $2^1$

8 is approximately $2^3$.

16 = $2^4$

Let's consider the differences between the exponents:

1-0 = 1

3-1 = 2

4-3 = 1

These don't seem to follow a clear pattern.

Another possible pattern is:

Multiply each term by successive powers of 2:

1 * 2 = 2

2 * 4 = 8

8 * 2 = 16

This also doesn't follow a clear pattern.

Let's examine differences between consecutive terms:

2-1 = 1

8-2 = 6

16-8 = 8

These differences don't seem to follow a clear pattern either.

However, let's consider the differences of the differences:

6-1 = 5

8-6 = 2

Again, no clear pattern is seen.

Alternatively, consider the sequence:

1, 2, 8, 16

Let's look at ratios of consecutive terms.

$r_1 = 2/1 = 2$

$r_2 = 8/2 = 4$

$r_3 = 16/8 = 2$

There seems to be no discernable pattern here.

The most appropriate way to find the next three terms is to consider the sequence as a function $f(n)$.

If we observe the terms carefully, we can rewrite the sequence as follows:

$a_1 = 1 = 1^1$

$a_2 = 2 = 2^1$

$a_3 = 8$ approximately $3^2$

$a_4 = 16$ is $4^2$

This sequence has no logical continuation. It appears that no reasonable solution can be found.

If we assume that the sequence is a typo, and instead is 1, 2, 4, 8, 16, ... which is $2^n$,

Then the terms are $2^0, 2^1, 2^2, 2^3, 2^4, ...$

The next three terms are $2^5, 2^6, 2^7$

Which gives 32, 64,

1

2

8.

iii) 10, 50, 250, 1250, ...

The ratio between consecutive terms are:

50/10 = 5

250/50 = 5

1250/250 = 5

Since the ratio between consecutive terms is constant, this is a geometric sequence with a common ratio of

5. Therefore, the next three terms are:

1250 * 5 = 6250

6250 * 5 = 31250

31250 * 5 = 156250

So, the next three terms are 6250, 31250, and

1

5

6

2

5

0.

3. Final Answer

i) 36, 47, 58

ii) Assuming the sequence is a typo and it is a geometric sequence of powers of 2, we obtain 32, 64,

1

2

8. iii) 6250, 31250, 156250