To determine the next number in the sequence 10, 12, 17, 30, 56, 100, 167 using the method of successive differences, follow these steps:
1. Calculate the first-order differences:
We subtract each term in the sequence from the subsequent term.
[tex]\[
\begin{align*}
12 - 10 &= 2 \\
17 - 12 &= 5 \\
30 - 17 &= 13 \\
56 - 30 &= 26 \\
100 - 56 &= 44 \\
167 - 100 &= 67 \\
\end{align*}
\][/tex]
The first-order differences are: [tex]\(2, 5, 13, 26, 44, 67\)[/tex].
2. Calculate the second-order differences:
We now find the differences between the first-order differences.
[tex]\[
\begin{align*}
5 - 2 &= 3 \\
13 - 5 &= 8 \\
26 - 13 &= 13 \\
44 - 26 &= 18 \\
67 - 44 &= 23 \\
\end{align*}
\][/tex]
The second-order differences are: [tex]\(3, 8, 13, 18, 23\)[/tex].
3. Calculate the third-order differences:
We proceed by finding the differences between the second-order differences.
[tex]\[
\begin{align*}
8 - 3 &= 5 \\
13 - 8 &= 5 \\
18 - 13 &= 5 \\
23 - 18 &= 5 \\
\end{align*}
\][/tex]
The third-order differences are: [tex]\(5, 5, 5, 5\)[/tex].
4. Determine the next third-order difference:
Since the third-order differences are constant and equal to 5, the next third-order difference remains [tex]\(5\)[/tex].
5. Determine the next second-order difference:
To find the next second-order difference, we add the last third-order difference of 5 to the last computed second-order difference of 23.
[tex]\[
23 + 5 = 28
\][/tex]
6. Determine the next first-order difference:
Similarly, we add the next second-order difference of 28 to the last first-order difference of 67.
[tex]\[
67 + 28 = 95
\][/tex]
7. Determine the next number in the sequence:
Finally, add the next first-order difference of 95 to the last term in the sequence, which is 167.
[tex]\[
167 + 95 = 262
\][/tex]
Therefore, the next number in the sequence is [tex]\(262\)[/tex].
