Question

1. what is a possible next term of this sequence 1, 0.25, 0.111..., 0.625, 0.04?

2. can you find two numbers that can satisfy this ? 6, 24, 60, 120, 210, ___, ___,

Answer 1

1. 0.0625
2. 336, 504

Explanation:

You want the next terms of the sequences given:

1. 1, 0.25, 0.111...
2. 6, 24, 60, 120, 210, __, __

1. Rationals

We can write the first sequence as the rational numbers ...

1/1, 1/4, 1/9, ...

We see that the denominators are the sequence of squares. This tells us the next term is 1/16 = 0.0625.

next term: 0.0625

2. Cubes

The first differences of the second sequence are ...

18 = 24-6, 36, 60, 90

The second differences are ...

18 = 36-18, 24, 30

The third differences are ...

6 = 24-18, 6

The constant 3rd differences tell us the sequence is a 3rd-degree sequence.

The corresponding sequence of cubes is ...

2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512

Comparing these to the given sequence terms, we find the differences to be 2, 3, 4, 5, 6. This suggests that the next two terms will be ...

7³ -7 = 336

8³ -8 = 504

The next two terms of the sequence are 336 and 504.

__

Additional comment

We assume that "0.625" in the question text is an answer choice that is supposed to be 0.0625 = 1/16.

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