Sequence Pattern Analysis

The question asks us to find the next term in the given number sequence: 64, 32, 48, 120, 420, ______. To solve this, we need to identify the pattern or rule that governs the progression of numbers in the sequence.

Step-by-Step Calculation

Let's examine the relationship between consecutive terms by looking at the multiplier needed to get from one term to the next:

• From the first term (64) to the second term (32): The multiplier is $\frac{32}{64} = 0.5$ or $\frac{1}{2}$. $64 \times \frac{1}{2} = 32$
• From the second term (32) to the third term (48): The multiplier is $\frac{48}{32} = 1.5$ or $\frac{3}{2}$. $32 \times \frac{3}{2} = 48$
• From the third term (48) to the fourth term (120): The multiplier is $\frac{120}{48} = 2.5$ or $\frac{5}{2}$. $48 \times \frac{5}{2} = 120$
• From the fourth term (120) to the fifth term (420): The multiplier is $\frac{420}{120} = 3.5$ or $\frac{7}{2}$. $120 \times \frac{7}{2} = 420$

Observing the sequence of multipliers (0.5, 1.5, 2.5, 3.5), we can see a clear pattern. Each multiplier is exactly 1 greater than the previous one. The multipliers can be written as fractions with a denominator of 2:

• First multiplier: $\frac{1}{2}$
• Second multiplier: $\frac{3}{2}$
• Third multiplier: $\frac{5}{2}$
• Fourth multiplier: $\frac{7}{2}$

The numerators (1, 3, 5, 7) are consecutive odd numbers. The sequence of multipliers is $\frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{7}{2}$.

Determining the Next Term

Following this identified pattern, the next multiplier should be derived from the next odd number, which is 9. So, the next multiplier will be $\frac{9}{2}$, which is equal to 4.5.

To find the next term in the sequence, we multiply the last given term (420) by this next multiplier (4.5):

Next Term $= 420 \times 4.5$

Let's perform the calculation:

Next Term $= 420 \times \frac{9}{2}$

Next Term $= (\frac{420}{2}) \times 9$

Next Term $= 210 \times 9$

Next Term $= 1890$

Therefore, the next term in the sequence 64, 32, 48, 120, 420, is 1890.