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Which of the following form of an AP ? J...

Which of the following form of an AP ? Justify Your answer.
(i) `-1,-1,-1,-1, …`
(ii) `0, 2, 0, 2, …`
(iii) `1, 1, 2, 2, 3, 3, … `
(iv) `11, 22, 33, … `
(v) `2, 2^(2), 2^(3), 2^(4)`

Text Solution

Verified by Experts

(i) Here, `t_(1)=1, t_(2)= -1, t_(3)= -1` and `t_(4)= -1`
`t_(2)-t_(1)= -1+1=0`
`t_(3)-t_(2)= -1+1=0`
`t_(4)-t_(3)= -1+1=0`
Clearly, the difference of successive terms is same, therefore given list of numbers form an AP.
(ii) Here, `t_(1)=0, t_(2)= 2, t_(3)= 0` and `t_(4)= 2`
`t_(2)-t_(1)= 2-0=2`
`t_(3)-t_(2)= 0-2= -2`
`t_(4)-t_(3)= 2-0=2`
Clearly, the difference of successive terms is not same, therefore given list of numbers does not form an AP.
(iii) Here, `t_(1)=1, t_(2)= 1, t_(3)= 2` and `t_(4)= 2`
`t_(2)-t_(1)= 1-1=0`
`t_(3)-t_(2)= 2-1=1`
`t_(4)-t_(3)= 2-2=0`
Clearly, the difference of successive terms is not same, therefore given list of numbers does not form an AP.
(iv) Here, `t_(1)=11, t_(2)= 22` and `t_(3)= 33`
`t_(2)-t_(1)= 22-11=11`
`t_(3)-t_(2)= 33-22=11`
`t_(4)-t_(3)= 33-22=11`
Clearly, the difference of successive terms is same, therefore given list of numbers form an AP.
(v) Here, `t_(1)=(1)/(2), t_(2)= (1)/(3)` and `t_(3)= (1)/(4)`
`t_(2)-t_(1)= (1)/(3)-(1)/(2)=(2-3)/(6)=(1)/(6)`
`t_(3)-t_(2)=(1)/(4)-(1)/(3)=(3-4)/(12)=(1)/(12)`
Clearly, the difference of successive terms is not same, therefore given list of numbers does not form an AP.
(vi) `2, 2^(2), 2^(3), 2^(4), ...` i.e.,`2, 4, 8, 16, ...`
Here, `t_(1)=2, t_(2)= 4, t_(3)= 8` and `t_(4)= 16`
`t_(2)-t_(1)= 4-2=2`
`t_(3)-t_(2)= 8-4= 4`
`t_(4)-t_(3)= 16-8=8`
Clearly, the difference of successive terms is not same, therefore given list of numbers does not form an AP.
(vii) `sqrt(3),sqrt(12),sqrt(27),sqrt(48), ...` i.e., `sqrt(3),2sqrt(3),3sqrt(3),4sqrt(3), ...`
Here, `t_(1)=sqrt(3), t_(2)= 2sqrt(3), t_(3)= 3sqrt(3)` and `t_(4)= 4sqrt(3)`
`t_(2)-t_(1)= 2sqrt(3)-sqrt(3)=sqrt(3)`
`t_(3)-t_(2)= 3sqrt(3)-2sqrt(3)=sqrt(3)`
`t_(4)-t_(3)= 4sqrt(3)-3sqrt(3)=sqrt(3)`
Clearly, the difference of successive terms is same, therefore given list of numbers form an AP.
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