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Question
If `(1 xx 2 + 2 xx 3 + 3 xx 4 + 4 xx 5 + ... "upto n terms")/(1 + 2 + 3 + 4 + ... "upto n terms") = 100/3,` find n
Solution
`(1 xx 2 + 2 xx 3 + 3 xx 4 + 4 xx 5 + ... "upto n terms")/(1 + 2 + 3 + 4 + ... "upto n terms") = 100/3,`
∴ `(sum_("r" = 1)^"n""r"("r" + 1))/(sum_("r" = 1)^"n" "r") = 100/3`
∴ `(sum_("r" = 1)^"n" "r"^2 + sum_("r" = 1)^"n" "r")/(sum_("r" = 1)^"n" "r") = 100/3`
∴ `(("n"("n" + 1)(2"n" + 1))/6 + ("n"("n" + 1))/2)/(("n"("n" + 1))/2) = 100/3`
∴ `(("n"("n" + 1))/6[(2"n" + 1) + 3])/(("n"("n" + 1))/2) = 100/3`
∴ `(2("n" + 2))/3= 100/3`
∴ n + 2 = 50
∴ n = 48
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