Write four more rational numbers in each of the following patterns:
(i) \(\frac{-3}{5},\frac{-6}{10},\frac{-9}{15},\frac{-12}{20}\)...
(ii) \(\frac{-1}{4},\frac{-2}{8},\frac{-3}{12},\)...
(iii) \(\frac{-1}{6},\frac{2}{-12},\frac{3}{-18},\frac{4}{-24}\),....
(iv)\(\frac{-2}{3},\frac{2}{-3},\frac{4}{-6},\frac{6}{-9}\),...
(i) \(\frac{-3}{5},\frac{-6}{10},\frac{-9}{15},\frac{-12}{20}\)...
(ii) \(\frac{-1}{4},\frac{-2}{8},\frac{-3}{12},\)...
(iii) \(\frac{-1}{6},\frac{2}{-12},\frac{3}{-18},\frac{4}{-24}\),....
(iv)\(\frac{-2}{3},\frac{2}{-3},\frac{4}{-6},\frac{6}{-9}\),...
Solution and Explanation
(i)\(\frac{-3}{5},\frac{-6}{10},\frac{-9}{15},\frac{-12}{20}\),...
= \(\frac{-3}{5},\frac{-3\times2}{5\times2},\frac{-3\times3}{5\times3},\frac{-3\times4}{5\times4}\),...
It can be observed that the numerator is a multiple of 3 while the denominator is a multiple of 5 and as we increase them further, these multiples are increasing. Therefore, the next four rational numbers in this pattern are
= \(\frac{-3\times5}{5\times5},\frac{-3\times6}{5\times6},\frac{-3\times7}{5\times7},\frac{-3\times8}{5\times8}\)...
= \(\frac{-15}{25},\frac{-18}{30},\frac{-21}{35},\frac{-24}{40}\)...
(ii) \(\frac{-1}{4},\frac{-2}{8},\frac{-3}{12},\)...
= \(\frac{-1}{4},\frac{-1\times2}{4\times2},\frac{-1\times3}{4\times3}.\)..
The next four rational numbers in this pattern are.
=\(\frac{-1\times4}{4\times4},\frac{-1\times5}{4\times5},\frac{-1\times6}{4\times6},\frac{-1\times7}{4\times7}.\)...
\(= \frac{-4}{16},\frac{-5}{20},\frac{-6}{24},\frac{-7}{28}...\)
(iii)\(\frac{-1}{6},\frac{2}{-12},\frac{3}{-18},\frac{4}{-24},....\)
\(= \frac{-1}{6},\frac{1\times2}{-6\times2},\frac{1\times3}{-6\times3},\frac{1\times4}{-6\times4}...\)
The next four rational numbers in this pattern are
\(\frac{1\times5}{-6\times5},\frac{1\times6}{-6\times6},\frac{1\times7}{-6\times7},\frac{1\times8}{-6\times8}....\)
\(\frac{5}{-30},\frac{6}{-36},\frac{7}{-42},\frac{8}{-48}...\)
(iv) \(\frac{-2}{3},\frac{2}{-3},\frac{4}{-6},\frac{6}{-9},...\)
\(\frac{-2}{3},\frac{2}{-3},\frac{2\times2}{-3\times2},\frac{2\times3}{-3\times3}...\)
The next four rational numbers in this pattern are
\(\frac{2\times4}{-3\times4},\frac{2\times5}{-3\times5},\frac{2\times6}{-3\times6},\frac{2\times7}{-3\times7}...\)
\(\frac{8}{-12},\frac{10}{-15},\frac{12}{-18},\frac{14}{-21}....\)
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