If for an Arithmetic Progression (AP),9 times | vedclass

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(A) Let the first term be $a$ and the common difference be $d$ for the given $AP$.The $n^{th}$ term of an $AP$ is given by $a_n = a + (n-1)d$.Given th

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(A) Let the first term be $a$ and the common difference be $d$ for the given $AP$.The $n^{th}$ term of an $AP$ is given by $a_n = a + (n-1)d$.Given that $9 	imes a_9 = 13 	imes a_{13}$.Substituting the formula for the terms:$9[a + (9-1)d] = 13[a + (13-1)d]$$9[a + 8d] = 13[a + 12d]$$9a + 72d = 13a + 156d$Rearranging the terms:$13a - 9a + 156d - 72d = 0$$4a + 84d = 0$Dividing by $4$:$a + 21d = 0$Since the $22^{nd}$ term $a_{22} = a + (22-1)d = a + 21d$,Therefore,$a_{22} = 0$.

If for an Arithmetic Progression $(AP)$,$9$ times the $9^{th}$ term is equal to $13$ times the $13^{th}$ term,then the value of the $22^{nd}$ term is:

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