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  • Which of the following form an AP? Justify your answer. –1, –1, –1, –1, ...

i)Which of the following form an AP? Justify your answer.

–1, –1, –1, –1, ...

ii) 

Which of the following form an AP? Justify your answer.

 0, 2, 0, 2, ...

iii)

Which of the following form an AP? Justify your answer.

1, 1, 2, 2, 3, 3,...

iv)

Which of the following form an AP? Justify your answer.

  11, 22, 33,...

v)

Which of the following form an AP? Justify your answer.

\frac{1}{2}, \frac{1}{3}, \frac{1}{4},......................

vi)

Which of the following form an AP? Justify your answer.

2, 2^2, 2^3, 2^4, ...

vii)

Which of the following form an AP? Justify your answer.

\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \ldots

Answers (1)

i)

Answer.   [True]

Solution.      

  \\-1, -1, -1, -1 ... $here a\textsubscript{1 }= -1 a\textsubscript{2 }= -1 a\textsubscript{3 }= -1 a\textsubscript{4}= -1\\ a\textsubscript{2} - a\textsubscript{1} = - 1 - (-1) = - 1 + 1 = 0\\ a\textsubscript{3} - a\textsubscript{2} = + 1(-1) = -1 + 1 = 0\\ a\textsubscript{4 }- a\textsubscript{3} = - 1 - (-1) = - 1 + 1 = 0\\

Here the difference of the successive term is same hence it is an AP.

ii)

Answer.          [No]

Solution.        

\\$The given series is 0, 2, 0, 2 ... here a\textsubscript{1 }= 0 a\textsubscript{2 }= 2 a\textsubscript{3 }= 0 a\textsubscript{4}= 2 \\ a\textsubscript{2} - a\textsubscript{1} = 2 - 0 = 2\\ a\textsubscript{3} - a\textsubscript{2} = 0 - 2 = - 2\\ a\textsubscript{4} - a\textsubscript{3} = 2 - 0 = 2\\

Here the difference successive term is not same. Hence it is not an AP.

iii)

Answer.          [No]

Solution.        

\\$The given series is {1, 1, 2, 2, 3, 3,...} here a\textsubscript{1 }= 1, a\textsubscript{2 }= 1, a\textsubscript{3 }= 2, a\textsubscript{4}= 2\\ a\textsubscript{2} - a\textsubscript{1} = 1 - 1 = 0\\ a\textsubscript{3} - a\textsubscript{2} = 2 - 1 = - 1\\ a\textsubscript{4} - a\textsubscript{3} = 2 - 2 = 0\\

 

Here the difference successive term is not the same. Hence it is not an AP.

iv)

Answer.          [Yes]

Solution.       

\\$The given series is {11, 22, 33,...} here a\textsubscript{1 }= 11, a\textsubscript{2 }= 22, a\textsubscript{3 }= 33\\ a\textsubscript{2} - a\textsubscript{1} = 22 - 11 = 11\\ a\textsubscript{3} - a\textsubscript{2} = 33 - 22 = 11\\

 

Here the difference of the successive term is same. Hence it is an AP.

v)

Answer.          [No]

Solution.        

\begin{aligned} &\text { Here the given series is } \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \ldots \ldots \text { here } \mathrm{a}_{1}=\frac{1}{2} \mathrm{a}_{2}=\frac{1}{3} \mathrm{a}_{3}=\frac{1}{4}\\ &\mathrm{a}_{2}-\mathrm{a}_{1}=\frac{1}{3}-\frac{1}{2}=\frac{2-3}{6}=\frac{-1}{6}\\ &a_{3}-a_{2}=\frac{1}{4}-\frac{1}{3}=\frac{3-4}{12}=\frac{-1}{12} \end{aligned}

Here the difference between the successive terms is not the same. Hence it is not an AP.

vi)

Answer.          [No]

Solution.        

\\$Here the given series is 2, 2\textsuperscript{2}, 2\textsuperscript{3}, 2\textsuperscript{4} $ \ldots $ .here a\textsubscript{1 }= 2 a\textsubscript{2 }= 2\textsuperscript{2 },a\textsubscript{3 }= 2\textsuperscript{3 } a\textsubscript{4}= 2\textsuperscript{4}\\ a\textsubscript{2} - a\textsubscript{1} = 2\textsuperscript{2} - 2 = 4 - 2 = 2\\ a\textsubscript{3} - a\textsubscript{2} = 2\textsuperscript{3} - 2\textsuperscript{2} = 8 - 4 = 4\\ a\textsubscript{4} - a\textsubscript{3} = 2\textsuperscript{4} - 2\textsuperscript{3} = 16 - 8 = 8\\

 

Here the difference of the successive terms is not the same. Hence it is not an AP.

vii)

Answer.          [Yes]

Solution.        

\begin{aligned} &\text { Here the given series is } \sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \ldots . \text { here } \mathrm{a}_{1}=\sqrt{3} \mathrm{a}_{2}=\sqrt{12} \mathrm{a}_{3}=\sqrt{27}\\ &a_{4}=\sqrt{48}\\ &a_{2}-a_{1}=\sqrt{12}-\sqrt{3}=2 \sqrt{3}-\sqrt{3}=\sqrt{3}=\sqrt{3}(2-1)=\sqrt{3}\\ &a_{3}-a_{2}=\sqrt{27}-\sqrt{12}=3 \sqrt{3}-2 \sqrt{3}=\sqrt{3}\\ &a_{4}-a_{3}=\sqrt{48}-\sqrt{27}=4 \sqrt{3}-3 \sqrt{3}=\sqrt{3} \end{aligned}

Here the difference of the successive term is the same. Hence it is an AP.

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