$1^{2}, 5^{2}, 7^{2}, 73, \ldots$ are $APs$? If they form an $AP,$ find the common difference $d$ and write three more terms.
(A) The given sequence is $1^{2}, 5^{2}, 7^{2}, 73, \ldots$
Calculating the values,we get $1, 25, 49, 73, \ldots$
To check if it is an $AP$,we find the difference between consecutive terms:
$a_{2} - a_{1} = 25 - 1 = 24$
$a_{3} - a_{2} = 49 - 25 = 24$
$a_{4} - a_{3} = 73 - 49 = 24$
Since the difference $a_{k+1} - a_{k}$ is constant $(d = 24)$,the given sequence forms an $AP$.
The common difference is $d = 24$.
The next three terms are:
$a_{5} = 73 + 24 = 97$
$a_{6} = 97 + 24 = 121$
$a_{7} = 121 + 24 = 145$
Calculating the values,we get $1, 25, 49, 73, \ldots$
To check if it is an $AP$,we find the difference between consecutive terms:
$a_{2} - a_{1} = 25 - 1 = 24$
$a_{3} - a_{2} = 49 - 25 = 24$
$a_{4} - a_{3} = 73 - 49 = 24$
Since the difference $a_{k+1} - a_{k}$ is constant $(d = 24)$,the given sequence forms an $AP$.
The common difference is $d = 24$.
The next three terms are:
$a_{5} = 73 + 24 = 97$
$a_{6} = 97 + 24 = 121$
$a_{7} = 121 + 24 = 145$
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