Find the next three terms of the sequence 1, 3, 7, 15, ...

Understand the Problem

The question is asking us to find the next three terms in the sequence 1, 3, 7, 15, and so on. This involves identifying the pattern in the sequence and applying it to extend the list of numbers.

Answer

The next terms are 31, 63, and 127.

Steps to Solve

Identify the Pattern in the Sequence
We start with the sequence given: 1, 3, 7, 15.
To find the pattern, we can observe the differences between each consecutive term:

$3 - 1 = 2$
$7 - 3 = 4$
$15 - 7 = 8$

The differences are 2, 4, and 8.

Observe the Differences Further
Next, let's see the differences between these differences:

$4 - 2 = 2$
$8 - 4 = 4$

The second set of differences is 2 and 4, which also doubles.

Extrapolate to Find the Next Difference
Since we're doubling each difference, the next difference after 8 would be:
$$ 8 \times 2 = 16 $$
Calculate the Next Terms
Now we add this new difference to the last term in the sequence:

Next term: $15 + 16 = 31$
Continuing this pattern, we will find the differences for the next terms:
After 16, the difference should again double:
$$ 16 \times 2 = 32 $$
Adding this gives: $31 + 32 = 63$
Next, we double 32 again:
$$ 32 \times 2 = 64 $$
Finally, add this to the last term: $63 + 64 = 127$

Thus the next three terms are 31, 63, and 127.

The next three terms in the sequence are 31, 63, and 127.

More Information

This sequence demonstrates a pattern where the differences between terms double each time, showcasing a form of exponential growth in the sequence. Such sequences can often be linked to binary representations or powers of two, reflecting their rapid increase.

Tips

Misunderstanding the Pattern: Some might assume a linear increase without recognizing the doubling of differences. To avoid this, carefully compute the differences iteratively.
Skipping Steps: Those unfamiliar with sequences may try to jump straight to a formula without calculating the differences. Taking each step systematically is crucial.

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