To determine the next number in the series [tex]\(4, 5, 7, 11, 19, \ldots\)[/tex], let's first analyze the pattern of differences between consecutive terms.
1. Calculate the differences between consecutive terms:
[tex]\[ 5 - 4 = 1 \][/tex]
[tex]\[ 7 - 5 = 2 \][/tex]
[tex]\[ 11 - 7 = 4 \][/tex]
[tex]\[ 19 - 11 = 8 \][/tex]
2. Observe the pattern in the differences:
[tex]\[ 1, 2, 4, 8 \][/tex]
These differences follow a pattern of doubling each time:
[tex]\[ 1 = 2^0 \][/tex]
[tex]\[ 2 = 2^1 \][/tex]
[tex]\[ 4 = 2^2 \][/tex]
[tex]\[ 8 = 2^3 \][/tex]
3. Given that the differences are powers of 2, the next difference should continue this pattern. The next power in this sequence is:
[tex]\[ 2^4 = 16 \][/tex]
4. To find the next term in the series, add this next difference to the last known term in the series:
[tex]\[ 19 + 16 = 35 \][/tex]
Therefore, the next number in the series is [tex]\( 35 \)[/tex].
