Determine whether the following sequence is an $A.P.$ or not. (Assume that the pattern continues.) If it is an $A.P.$,find its $n^{th}$ term: $5, 11, 17, 23, \ldots$
(A) To determine if the sequence $5, 11, 17, 23, \ldots$ is an $A.P.$,we check the common difference $d$ between consecutive terms.
$d_1 = 11 - 5 = 6$
$d_2 = 17 - 11 = 6$
$d_3 = 23 - 17 = 6$
Since the common difference $d = 6$ is constant,the sequence is an $A.P.$
The formula for the $n^{th}$ term of an $A.P.$ is $T_n = a + (n - 1)d$,where $a = 5$ and $d = 6$.
$T_n = 5 + (n - 1)6$
$T_n = 5 + 6n - 6$
$T_n = 6n - 1$
$d_1 = 11 - 5 = 6$
$d_2 = 17 - 11 = 6$
$d_3 = 23 - 17 = 6$
Since the common difference $d = 6$ is constant,the sequence is an $A.P.$
The formula for the $n^{th}$ term of an $A.P.$ is $T_n = a + (n - 1)d$,where $a = 5$ and $d = 6$.
$T_n = 5 + (n - 1)6$
$T_n = 5 + 6n - 6$
$T_n = 6n - 1$
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