Are $a, a^{2}, a^{3}, a^{4}, \ldots$ in an $AP$? If they form an $AP$,find the common difference $d$ and write three more terms.
(NONE) The given sequence is $a, a^{2}, a^{3}, a^{4}, \ldots$
To check if the sequence is in an $AP$,we calculate the difference between consecutive terms:
$a_{2} - a_{1} = a^{2} - a = a(a - 1)$
$a_{3} - a_{2} = a^{3} - a^{2} = a^{2}(a - 1)$
$a_{4} - a_{3} = a^{4} - a^{3} = a^{3}(a - 1)$
Since the difference $a_{k+1} - a_{k}$ is not constant (i.e.,$a(a-1) \neq a^{2}(a-1)$ for $a \neq 0, 1$),the given sequence does not form an $AP$.
To check if the sequence is in an $AP$,we calculate the difference between consecutive terms:
$a_{2} - a_{1} = a^{2} - a = a(a - 1)$
$a_{3} - a_{2} = a^{3} - a^{2} = a^{2}(a - 1)$
$a_{4} - a_{3} = a^{4} - a^{3} = a^{3}(a - 1)$
Since the difference $a_{k+1} - a_{k}$ is not constant (i.e.,$a(a-1) \neq a^{2}(a-1)$ for $a \neq 0, 1$),the given sequence does not form an $AP$.
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