Question

Show that each of the following sequences is an A.P. Also find the common difference and write 3 more terms in each case.
(i) 3, −1, −5, −9 ...
(ii) −1, 1/4, 3/2, 11/4, ...
(iii) $\sqrt{2},3\sqrt{2},5\sqrt{2},7\sqrt{2},...$
(iv) 9, 7, 5, 3, ...

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{We}\mathrm{have}: \\ -1-3=-4, \\ -5-(-1)=-4, \\ -9-(-5)=-4... \\ \mathrm{Thus},\mathrm{the}\mathrm{sequence}\mathrm{is}\mathrm{an}\mathrm{A}.\mathrm{P}.\mathrm{with}\mathrm{the}\mathrm{common}\mathrm{difference}\mathrm{being}-4. \\ \mathrm{The}\mathrm{next}\mathrm{three}\mathrm{terms}\mathrm{are}\mathrm{as}\mathrm{follows}: \\ -9-4=-13 \\ -13-4=-17 \\ -17-4=-21 \\ \left(\mathrm{ii}\right)\mathrm{We}\mathrm{have}: \\ 1/4-(-1)=5/4 \\ 3/2-1/4=5/4 \\ 11/4-3/2=5/4 \\ \mathrm{Thus},\mathrm{the}\mathrm{sequence}\mathrm{is}\mathrm{an}\mathrm{A}.\mathrm{P}.\mathrm{with}\mathrm{the}\mathrm{common}\mathrm{difference}\mathrm{being}(5/4). \\ \mathrm{The}\mathrm{next}\mathrm{three}\mathrm{terms}\mathrm{are}\mathrm{as}\mathrm{follows}: \\ 11/4+5/4=16/4=4 \\ 16/4+5/4=21/4 \\ 21/4+5/4=26/4 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{We}\mathrm{have}: \\ 3\sqrt{2}-\sqrt{2}=2\sqrt{2} \\ 5\sqrt{2}-3\sqrt{2}=2\sqrt{2} \\ 7\sqrt{2}-5\sqrt{2}=2\sqrt{2} \\ \mathrm{Thus},\mathrm{the}\mathrm{sequence}\mathrm{is}\mathrm{an}\mathrm{A}.\mathrm{P}.\mathrm{with}\mathrm{the}\mathrm{common}\mathrm{difference}\mathrm{being}\left(2\sqrt{2}\right). \\ \mathrm{The}\mathrm{next}\mathrm{three}\mathrm{terms}\mathrm{are}\mathrm{as}\mathrm{follows}: \\ 7\sqrt{2}+2\sqrt{2}=9\sqrt{2} \\ 9\sqrt{2}+2\sqrt{2}=11\sqrt{2} \\ 11\sqrt{2}+2\sqrt{2}=13\sqrt{2} \\ \left(\mathrm{iv}\right)\mathrm{We}\mathrm{have}: \\ 7-9=-2 \\ 5-7=-2 \\ 3-5=-2 \\ \mathrm{Thus},\mathrm{the}\mathrm{sequence}\mathrm{is}\mathrm{an}\mathrm{A}.\mathrm{P}.\mathrm{with}\mathrm{the}\mathrm{common}\mathrm{difference}\mathrm{being}(-2). \\ \mathrm{The}\mathrm{next}\mathrm{three}\mathrm{terms}\mathrm{are}\mathrm{as}\mathrm{follows}: \\ 3-2=1 \\ 1-2=-1 \\ -1-2=-3 \end{gathered}$$