To determine the next number in the series given by [tex]\(67, 57, 59, 63\)[/tex], let's analyze the pattern and deduce the logical progression step by step.
1. Identify the sequence:
- The given numbers are 67, 57, 59, and 63.
2. Observe the differences between consecutive numbers:
- From 67 to 57, the change is [tex]\( 57 - 67 = -10 \)[/tex].
- From 57 to 59, the change is [tex]\( 59 - 57 = +2 \)[/tex].
- From 59 to 63, the change is [tex]\( 63 - 59 = +4 \)[/tex].
3. Identify the increments in the differences:
- From -10 to +2, there is an increment of [tex]\( +12 \)[/tex] (since [tex]\(-10 + 12 = 2\)[/tex]).
- From +2 to +4, there is an increment of [tex]\( +2 \)[/tex].
4. Assume the next increment:
- Continuing the pattern, the difference increment has been increasing by a certain pattern. Initially, it jumped by 12, and then by 2. To follow an orderly pattern, let’s consider that the next increment continues by a fixed step. The step appears to increase by 2 each time: [tex]\(10 (although it was initially 12 jump) → 2 → 4 → \dots \)[/tex]
5. Calculate the next difference:
- The last difference was +4. If we continue increasing the step by 2, the next should be [tex]\(+6\)[/tex].
6. Add this difference to the last number in the series:
- The last number given is 63. Adding the next difference of +6 to 63 results in [tex]\( 63 + 6 = 69 \)[/tex].
Therefore, the next number in the series is [tex]\( \boxed{69} \)[/tex].
