If the sum of the first 14 terms of an AP is | vedclass

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(D) Given: Sum of first $14$ terms $S_{14} = 1050$,first term $a = 10$,and number of terms $n = 14$.The formula for the sum of the first $n$ terms of

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$200$

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(D) Given: Sum of first $14$ terms $S_{14} = 1050$,first term $a = 10$,and number of terms $n = 14$.The formula for the sum of the first $n$ terms of an $AP$ is $S_n = \frac{n}{2}[2a + (n - 1)d]$.Substituting the values: $1050 = \frac{14}{2}[2(10) + (14 - 1)d]$.$1050 = 7[20 + 13d]$.$150 = 20 + 13d$.$130 = 13d$,which gives $d = 10$.The $n^{th}$ term of an $AP$ is given by $a_n = a + (n - 1)d$.For the $20^{th}$ term: $a_{20} = 10 + (20 - 1) 	imes 10$.$a_{20} = 10 + 19 	imes 10 = 10 + 190 = 200$.Thus,the $20^{th}$ term is $200$.

$S.No.$	$a$	$d$	$n$	$a_{n}$
$(i)$	$7$	$3$	$8$	$...$
$(ii)$	$-18$	$...$	$10$	$0$
$(iii)$	$...$	$-3$	$18$	$-5$
$(iv)$	$-18.9$	$2.5$	$...$	$3.6$
$(v)$	$3.5$	$0$	$105$	$...$

If the sum of the first $14$ terms of an $AP$ is $1050$ and its first term is $10,$ find the $20^{th}$ term.

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