Question
What number comes next in the series? \( 1,2,6,15,31 \)
Ask by Boyd Marshall. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The next number in the series is 56.
Solution
The sequence is given as
\[
1, \quad 2, \quad 6, \quad 15, \quad 31.
\]
1. First, compute the differences between consecutive terms:
\[
2 - 1 = 1, \quad 6 - 2 = 4, \quad 15 - 6 = 9, \quad 31 - 15 = 16.
\]
2. Notice that the differences \(1, 4, 9,\) and \(16\) are perfect squares:
\[
1 = 1^2, \quad 4 = 2^2, \quad 9 = 3^2, \quad 16 = 4^2.
\]
3. The next difference should then be the square of the next integer:
\[
5^2 = 25.
\]
4. Add this difference to the last term in the series:
\[
31 + 25 = 56.
\]
Thus, the next number in the series is
\[
56.
\]
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Simplify this solution The Deep Dive
To find the next number in the series \( 1, 2, 6, 15, 31 \), let's examine the differences between consecutive terms: - \( 2 - 1 = 1 \) - \( 6 - 2 = 4 \) - \( 15 - 6 = 9 \) - \( 31 - 15 = 16 \) Now, look at these differences: \( 1, 4, 9, 16 \). These are \( 1^2, 2^2, 3^2, 4^2 \), suggesting the next difference should be \( 5^2 = 25 \). Adding this to the last term in the series: \[ 31 + 25 = 56 \] Thus, the next number in the series is \( 56 \).
