To determine the next term in the sequence [tex]\( 25, 10, 4, \ldots \)[/tex], we need to identify the pattern or rule governing the sequence.
1. Identify the first three terms in the sequence and their differences:
- The first term ([tex]\( T_1 \)[/tex]) is 25.
- The second term ([tex]\( T_2 \)[/tex]) is 10.
- The third term ([tex]\( T_3 \)[/tex]) is 4.
2. Calculate the differences between consecutive terms:
- The difference between the second and the first term ([tex]\( T_2 - T_1 \)[/tex]) is [tex]\( 10 - 25 = -15 \)[/tex].
- The difference between the third and the second term ([tex]\( T_3 - T_2 \)[/tex]) is [tex]\( 4 - 10 = -6 \)[/tex].
3. Examine the change in the differences:
- Calculate the change (or difference) between the consecutive differences to identify a pattern.
- The change in the differences ([tex]\( \Delta_2 - \Delta_1 \)[/tex]) is [tex]\( -6 - (-15) = -6 + 15 = 9 \)[/tex].
4. Predict the difference for the next term:
- Assuming that our sequence follows a consistent rule, we add the constant change observed to the most recent difference.
- Therefore, the next difference to the third term should be [tex]\( -6 + 9 = 3 \)[/tex].
5. Calculate the next term:
- The next term will be the third term plus this newly calculated difference.
- The next term is [tex]\( 4 + 3 = 7 \)[/tex].
Thus, the next term in the sequence [tex]\( 25, 10, 4, \ldots \)[/tex] is [tex]\( 7 \)[/tex]. The detailed step-by-step solution confirms that the next term is 7.
