Answer:
.
Sequence 7: \(-22, -10, -4, -1, -\frac{1}{2}, \ldots\)
Identify the pattern:
[tex]{- \text{Difference between} \(-22\) and \(-10\): \(12\)}[/tex]
[tex]{- \text{Difference between} \(-10\) and \(-4\): \(6\)}[/tex]
[tex]{- \text{Difference between} \(-4\) and \(-1\): \(3\)}[/tex]
[tex]{ \text{Difference between} \(-1\) and \(-\frac{1}{2}\): \(\frac{1}{2}\)}[/tex]
Next term:
[tex]- \text{Next difference}: \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\)[/tex]
[tex]- \text{Next term}: \(-\frac{1}{2} + \frac{1}{4} = -\frac{1}{4}\)[/tex]
Following the same pattern:
[tex]- \text{Next difference}: \(\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}\)[/tex]
[tex]- \text{Next term}: \(-\frac{1}{4} + \frac{1}{8} = -\frac{1}{8}\)[/tex]
Further following the pattern:
[tex]- \text{Next difference}: \(\frac{1}{8} \times \frac{1}{2} = \frac{1}{16}\)[/tex]
[tex]- \text{Next term}: \(-\frac{1}{8} + \frac{1}{16} = -\frac{1}{16}\)[/tex]
So, the next three terms are:
[tex]-\frac{1}{4}, -\frac{1}{8}, -\frac{1}{16}[/tex]
#Sequence 8: \(-6, -3, -2, -\frac{3}{2}, -\frac{6}{5}, \ldots\)
Identify the pattern:
[tex]{- \text{Ratio between} \(-6\) and \(-3\): \(\frac{-3}{-6} = \frac{1}{2}\)}[/tex]
[tex]{- \text{Ratio between} \(-3\) and \(-2\): \(\frac{-2}{-3} \approx \frac{2}{3}\)}[/tex]
[tex]{- \text{Ratio between} \(-2\) and \(-\frac{3}{2}\): \(\frac{-\frac{3}{2}}{-2} = \frac{3}{4}\)}[/tex]
[tex]{- \text{Ratio between} \(-\frac{3}{2}\) and \(-\frac{6}{5}\): \(\frac{-\frac{6}{5}}{-\frac{3}{2}} = \frac{4}{5}\)}[/tex]
Ratios seem to be
[tex]\(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\), \text {indicating the next ratio will be} \(\frac{5}{6}\).[/tex]
Next term:
[tex]- \text{Next term}: \(-\frac{6}{5} \times \frac{5}{6} = -1\)[/tex]
Following the pattern:
[tex]- \text{Next ratio}: \(\frac{6}{7}\)[/tex]
[tex]- \text{Next term}: \(-1 \times \frac{6}{7} = -\frac{6}{7}\)[/tex]
Further following the pattern:
[tex]- \text{Next ratio}: \(\frac{7}{8}\)[/tex]
[tex]{ - \text{Next term}: \(-\frac{6}{7} \times \frac{7}{8} = -\frac{6}{8} = -\frac{3}{4}\)}[/tex]
So, the next three terms are:
[tex]-1, -\frac{6}{7}, -\frac{3}{4}[/tex]
Sequence 9: \(4, 16, 36, 64, 100, \ldots\)
Identify the pattern: These numbers are perfect squares.
[tex]\(4 = 2^2\)[/tex]
[tex]\(16 = 4^2\)[/tex]
[tex]\(36 = 6^2\)[/tex]
[tex](64 = 8^2\)[/tex]
[tex](100 = 10^2)[/tex]
Next terms:
[tex]- {Next \: term: \(12^2 = 144\)}[/tex]
[tex] {Following \: term: \(14^2 = 196\)}[/tex]
[tex]Further \: term: \(16^2 = 256\)[/tex]
So, the next three terms are:
[tex]144, 196, 256[/tex]
Sequence 10: \(26, 36, 46, 56, 66, \ldots\)
Identify the pattern: These numbers are increasing by (10).
Next terms:
[tex]- \text{Next term: \(66 + 10 = 76\)}[/tex]
[tex]- \text{Following term: \(76 + 10 = 86}[/tex]
