Understanding the Number Series Problem

The question asks us to find the next number in the given sequence: \(19, 37, 73, 145, ?\) To solve this, we need to identify the pattern governing the series.

Identifying the Pattern

Let's examine the relationship between consecutive numbers in the series:

• The first term is 19.
• The second term is 37.
• The third term is 73.
• The fourth term is 145.

We can look at the difference between terms:

• Difference between the second and first term: \(37 - 19 = 18\)
• Difference between the third and second term: \(73 - 37 = 36\)
• Difference between the fourth and third term: \(145 - 73 = 72\)

The differences are \(18, 36, 72\). We can observe that each difference is double the previous difference:

• \(18 \times 2 = 36\)
• \(36 \times 2 = 72\)

Following this pattern, the next difference should be \(72 \times 2 = 144\).

Calculating the Next Number

To find the next number in the series, we add the calculated next difference (144) to the last known term (145):

Next Number = Last Term + Next Difference

Next Number = \(145 + 144\)

Next Number = \(289\)

Alternative Pattern Verification

Another way to look at the pattern is by multiplying the current term by 2 and subtracting 1:

• \(19 \times 2 - 1 = 38 - 1 = 37\)
• \(37 \times 2 - 1 = 74 - 1 = 73\)
• \(73 \times 2 - 1 = 146 - 1 = 145\)

Applying this rule to find the next term:

• Next Term = \(145 \times 2 - 1\)
• Next Term = \(290 - 1\)
• Next Term = \(289\)

Both methods confirm that the next number in the series is 289.

Conclusion

The pattern identified is consistent, and the calculation shows that the next number in the series \(19, 37, 73, 145, ?\) is 289.