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Which of the following are A.P's ? If t...

Which of the following are A.P's ? If they form an A.P., find the common difference 'd' and write two more terms :
(i) 2, 4, 8, 16, ... (ii) `2, (5)/(2), 3, (7)/(2), ...`
(iii) -1.2, -3.2, -5.2, -7.2, .... (iv) `-(1)/(2),-(1)/(2),-(1)/(2),-(1)/(2),...`

Text Solution

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(i) 2, 4, 8, 16, ...
Here, `a_(2)-a_(1)=4-2=2`
`a_(3)-a_(2)=8-4=4`
`:. a_(2)-a_(1)!= a_(3)-a_(2)`
`:. a_(n)-a_(n-1)` is not same every time.
(ii) `2, (5)/(2), 3, (7)/(2), ...`
Here, `a_(2)-a_(1)=(5)/(2)-2=(1)/(2)`
`a_(3)-a_(2)=3-(5)/(2)=(1)/(2)`
`a_(4)-a_(3)=(7)/(2)-3=(1)/(2)`
`:. a_(n)-a_(n-1)` is same every time.
`rArr` Given numbers form an A.P.
Now, common difference `d=(1)/(2)`
The next two terms are `(7)/(2)+(1)/(2)=4` and `4+(1)/(2)=(9)/(2)`.
(iii) -1.2, -3.2, -5.2, -7.2, ...
Here, `a_(2)-a_(1)=(-3.2)-(-1.2)=-2`
`a_(3)-a_(2)=(-5.2)-(-3.2)=-2`
`a_(4)-a_(3)=(-7.2)-(-5.2)=-2`
`:. a_(n)-a_(n-1)` is same every time.
`rArr` Given numbers form an A.P.
Now, common difference d=-2
The next two numbers are `-7.2 + (-2)=-9.2 and (-9.2)+(-2)=-11.2`.
(iv) `-(1)/(2),-(1)/(2),-(1)/(2),-(1)/(2),...`
Here, `a_(2)-a_(1)=(-(1)/(1))-(-(1)/(2))=0`
`a_(3)-a_(2)=(-(1)/(2))-(-(1)/(2))=0`
`a_(4)-a_(3)=(-(1)/(2))-(-(1)/(2))=0`
`:. a_(n)-a_(n-1)` is same every time.
`rArr` Given numbers form an A.P.
Now, common difference d = 0
The next two terms are `-(1)/(2)+0=-(1)/(2)` and `-(1)/(2)+0=-(1)/(2)`.
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