If the 5^{th} and the 12^{th} terms of an A.P | vedclass

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(C) Let $a$ be the first term and $d$ be the common difference of an $A.P.$The $n^{th}$ term of an $A.P.$ is given by $a_n = a + (n-1)d$.Given $a_5 =

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$1150$

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(C) Let $a$ be the first term and $d$ be the common difference of an $A.P.$The $n^{th}$ term of an $A.P.$ is given by $a_n = a + (n-1)d$.Given $a_5 = 30$,we have $a + 4d = 30$  --- $(1)$Given $a_{12} = 65$,we have $a + 11d = 65$ --- $(2)$Subtracting equation $(1)$ from equation $(2)$:$(a + 11d) - (a + 4d) = 65 - 30$$7d = 35$$d = 5$Substituting $d = 5$ in equation $(1)$:$a + 4(5) = 30$$a + 20 = 30$$a = 10$The sum of the first $n$ terms is given by $S_n = \frac{n}{2}[2a + (n-1)d]$.For $n = 20$:$S_{20} = \frac{20}{2}[2(10) + (20-1)5]$$S_{20} = 10[20 + 19 	imes 5]$$S_{20} = 10[20 + 95]$$S_{20} = 10 	imes 115 = 1150$.

If the $5^{th}$ and the $12^{th}$ terms of an $A.P.$ are $30$ and $65,$ respectively,then what is the sum of the first $20$ terms?

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