(i) The given progression 9, 15, 21, 27,…………
Clearly, 15 – 9 = 21 – 15 = 27 – 21 = 6 (Constant)
Thus, each term differs from its preceding term by 6.
So, the given progression is an AP.
First term = 9
Common difference = 6
Next term of the AP = 27 + 6 = 33
(ii) The given progression 11, 6, 1, – 4,……..
Clearly, 6 – 11 = 1 – 6 = –4 – 1 = –5 (Constant)
Thus, each term differs from its preceding term by 6.
So, the given progression is an AP.
First term = 11
Common difference = –5
Next term of the \(Ap=-4(-5)=-9\)
(iii) The given progression \(-1,\frac{-5}{6},\frac{-2}{3},\frac{-1}{2},\)..........
Clearly, \(\frac{-5}{6}-(-1)=\frac{-2}{3}-(\frac{-5}{6})\) = \(\frac{-1}{2}-(\frac{-2}{3})=\frac{1}{6}\)(Constant)
Thus, each term differs from its preceding term by \(\frac{1}{6}.\)
So, the given progression is an AP.
First term = –1
Common difference = \(\frac{1}{6}\)
Next tern of the \(AP=\frac{-1}{2}+\frac{1}{6}=\frac{-2}{6}=\frac{-1}{3}\)
(iv) The given progression \(\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32},\)..........
This sequence can be written as \(\sqrt{2},2\sqrt{2},3\sqrt{2},4\sqrt{2},\)........
Thus, each term differs from its preceding term by \(\sqrt{2}\)
So, the given progression is an AP.
First term = \(\sqrt{2}\)
Common difference = \(\sqrt{2}\)
Next tern of the \(AP=4\sqrt{2}+\sqrt{2}=5\sqrt{2}=\sqrt{50}\)
(v) This given progression \(\sqrt{20},\sqrt{45},\sqrt{80},\sqrt{125},\).........
This sequence can be re-written as \(2\sqrt{5},3\sqrt{5},4\sqrt{5},5\sqrt{5},\)...........
Thus, each term differs from its preceding term by \(\sqrt{5}.\)
So, the given progression is an AP.
First term = \(2\sqrt{5}=\sqrt{20}\)
Common difference = \(\sqrt{5}\)
Next term of the AP \(5\sqrt{5}+\sqrt{5}=6\sqrt{5}=\sqrt{180}\)
