Find the sum of all integers between 50 and 5 | vedclass

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(D) The first integer after $50$ that is divisible by $7$ is $56$,and the last integer before $500$ that is divisible by $7$ is $497.$The sequence of

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$17696$

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(D) The first integer after $50$ that is divisible by $7$ is $56$,and the last integer before $500$ that is divisible by $7$ is $497.$The sequence of integers between $50$ and $500$ divisible by $7$ is $56, 63, 70, \ldots, 497.$This forms an Arithmetic Progression $(A.P.)$ where the first term $a = 56$ and the common difference $d = 7.$Using the formula for the $n^{th}$ term: $a_n = a + (n - 1)d.$Substituting the values: $497 = 56 + (n - 1) 	imes 7.$$497 - 56 = (n - 1) 	imes 7 \implies 441 = (n - 1) 	imes 7.$$n - 1 = 441 / 7 = 63 \implies n = 64.$The sum of an $A.P.$ is given by $S_n = \frac{n}{2}(a + a_n).$$S_{64} = \frac{64}{2}(56 + 497) = 32 	imes 553 = 17696.$

Find the sum of all integers between $50$ and $500$ which are divisible by $7.$

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