The Gateway to Computer Science Excellence
First time here? Checkout the FAQ!
x
+9 votes
775 views

Let $P =\sum_{\substack{1\le i \le 2k \\ i\;odd}} i$ and $Q = \sum_{\substack{1 \le i \le 2k \\ i\;even}} i$, where $k$ is a positive integer. Then

  1. $P = Q - k$
  2. $P = Q + k$
  3. $P = Q$
  4. $P = Q + 2k$
asked in Combinatory by Veteran (59.5k points)
retagged by | 775 views

3 Answers

+13 votes
Best answer
$\textbf{P}=1+3+5+7+.....+(2k-1)$

     $=(2-1)+(4-1)+(6-1)+(8-1)+.......+(2k-1)$

     $=(2+4+6+8+.....2k)+(-1-1-1-1-1.....k \  \text{times})$

     $=\textbf{Q}+(-k)=\textbf{Q-k}$
answered by Active (4.1k points)
edited by
+8 votes
Substitute k=3 then we get p=9 and q=12  on verifying we get option A.
answered by Active (1.2k points)
+9
That's true. P is adding the first k odd numbers and Q is adding the first k even numbers. Each even number is 1 greater than its corresponding odd number being added. So for k additions, Q will be P+k
0
but option c also satisfies given condition........................... checking for each k
0
@sanjay how option c will true please explain?
+5 votes
The odd series is 1 3 5 7 ... 2k-1

So, 1+(t1-1)2=2k-1

      or t1=k;

P = (k/2) [2x1+(k-1)2]=k^2

The even series is 2 4 6 8 10  ... 2k

So, 2+(t2-1)2=2k

      or t2=k;

Q = (k/2) [2x2+(k-1)2]=k^2+k
So P=Q-k  is answer...
answered by Active (1.5k points)
edited by
0

Nice approach.Thanks :-)

Just small correction needed."The odd series is 2 4 6 8 10  ... 2k ." 



Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true

38,235 questions
45,737 answers
133,029 comments
49,929 users