Integers 1,2,3 . n where n>2 , are written on a board. Two numbers m, k such that 1 < m < n .1 < k < n are removed and the average of the remaining numbers is found to be 17 . What is the maximum sum of the two removed

Correct answer: 51 aligned & ((n(n+1))/(2)-(2 n-1))/(n-2) 31 & n=32,33,34 & case- 1, n=32 aligned aligned & ((n(n+1))/(2)-p)/((n-2))=17 (n(n+1))/(2)-17(n-2)=p & p=18 & case- 2, n=33 p=34 & case- 3, n=34 p=51 & Maximum sum =51 aligned

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Sequences and Series

Olympiad workbook · IOQM · Sequences and Series

Integers 1,2,3 . n where n>2 , are written on a board. Two numbers...

Integers

1,2,3 \ldots \ldots . \mathrm{n}

where

\mathrm{n}>2

, are written on a board. Two numbers

\mathrm{m}, \mathrm{k}

such that

1 < \mathrm{m} < \mathrm{n} .1 < \mathrm{k} < \mathrm{n}

are removed and the average of the remaining numbers is found to be 17 . What is the maximum sum of the two removed numbers?

Correct answer

Step-by-step solution

\begin{aligned} & \frac{\frac{n(n+1)}{2}-(2 n-1)}{n-2} < 17 < \frac{\frac{n(n+1)}{2}-3}{n-2} \\ & \frac{n^2+n-4 n+2}{2(n-2)} < 17 < \frac{n^2+n-6}{2(n-2)} \\ & \frac{n^2+3 n+2}{2(n-2)} < 17 < \frac{(n+3)(n-2)}{2(n-2)} \\ & \frac{n-1}{2} < 17 < \frac{n+3}{2} \\ & n < 35 \text { and } n>31 \\ & n=32,33,34 \\ & \text { case- } 1, n=32\end{aligned}

\begin{aligned} & \frac{\frac{n(n+1)}{2}-p}{(n-2)}=17 \Rightarrow \frac{n(n+1)}{2}-17(n-2)=p \\ & p=18 \\ & \text { case- } 2, n=33 \Rightarrow p=34 \\ & \text { case- } 3, n=34 \Rightarrow p=51 \\ & \text { Maximum sum }=51\end{aligned}

Practice Sequences and Series on Quantrex Academy →

More from Sequences and Series

A finite set M of positive integers consists of distinct perfect squares and the number… Olympiad workbook If ∑_ k=1 ^N (2 k+1)/((k^2+k)^2)=0.9999 then determine the value of N . Olympiad workbook A group of women working together at the same rate can build a wall in 45 hours. When the… Olympiad workbook Let m be the smallest positive integer such that m^2+(m+1)^2+…+(m+10)^2 is the square of… Olympiad workbook Let u, v, w be real numbers in geometric progression such that u>v>w . Suppose u^ 40 =v^… Olympiad workbook The sequence a_n _ n ≥ 0 is defined by a_0=1, a_1=-4 and a_ n+2 =-4 a_ n+1 -7 a_n , for n… Olympiad workbook Let the sum ∑_ n=1 ^9 (1)/(n(n+1)(n+2)) written in its lowest terms be (p)/(q) . Find the… Olympiad workbook Consider the set T of all triangles whose sides are distinct prime numbers which are also… Olympiad workbook Full Sequences and Series list All Olympiad workbook PYQs