search
Log In
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
15 votes
2.3k views

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

  1. $P = Q - k$
  2. $P = Q + k$
  3. $P = Q$
  4. $P = Q + 2k$
in Combinatory
edited by
2.3k views

8 Answers

26 votes
 
Best answer
$\textbf{P}=1+3+5+7+\ldots +(2k-1)$
$\quad=(2-1)+(4-1)+(6-1)+(8-1)+\ldots+(2k-1)$
$\quad=(2+4+6+8+\ldots+2k)+(-1-1-1-1-1\ldots k \  \text{times})$
$\quad=\textbf{Q}+(-k)=\textbf{Q-k}$

Correct Answer: $A$

edited by
10 votes
Substitute k=3 then we get p=9 and q=12  on verifying we get option A.
11
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?
0
Lets Assume the value of k=5

then we take number from 1 to 10.

then P= 1+3+5+7+9 = 25

and Q= 2+4+6+8+10 = 30            So, here we conclude that P=Q+K
0
@SHIV_KANNAUJ

LoL
6 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...

edited by
0

Nice approach.Thanks :-)

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

0 votes
Assme k=3 so 2k=6

p-sum of all odd numbers from 1 to 2k so p=1+3+5=9

q-sum of all even numbers from 1 to 2k so q=2+4+6=12

now as we can see P=Q-K
0 votes
Just Write –

P = 1 + 3 + 5 + …

Q = 2 + 4 + 6 + …

and take K = 3

so P = (1+3+5) = 9

and Q = (2+4+6) = 12

and equation 1 holds-

P = Q – K

9 = 12 – 3.

it take only 30 second question to solve  :)
0 votes

$\therefore$ Answer is Option $\LARGE A$

0 votes
A
0 votes

Try to use substitution method here

Let k=2

 We know that 1<= i <= 2k

That is 1<=i<= 4

For P i is odd 

For Q i is even

Therefore P will be 1+3 => 4

Q will be 2+4 =>6

Now check options and substitute values of P , Q ,k in options

a) P = Q-k

4 = 6-2

4=4

b) P=Q+k

4 = 6+2

4 != 8

c) P=Q

4!= 6

d) P=Q+2k

4 = 6+2(2)

4!= 10

Only option a) is satisfying

Therefore correct answer is option A

ago
Answer:

Related questions

39 votes
4 answers
1
15.4k views
Which of the following is NOT true of deadlock prevention and deadlock avoidance schemes? In deadlock prevention, the request for resources is always granted if the resulting state is safe In deadlock avoidance, the request for resources is ... state is safe Deadlock avoidance is less restrictive than deadlock prevention Deadlock avoidance requires knowledge of resource requirements apriori..
asked Sep 12, 2014 in Operating System Kathleen 15.4k views
22 votes
2 answers
2
8.6k views
If a class $B$ network on the Internet has a subnet mask of $255.255.248.0$, what is the maximum number of hosts per subnet? $1022$ $1023$ $2046$ $2047$
asked Sep 12, 2014 in Computer Networks Kathleen 8.6k views
20 votes
4 answers
3
5.5k views
Which of the following statements is false? Every NFA can be converted to an equivalent DFA Every non-deterministic Turing machine can be converted to an equivalent deterministic Turing machine Every regular language is also a context-free language Every subset of a recursively enumerable set is recursive
asked Sep 12, 2014 in Theory of Computation Kathleen 5.5k views
39 votes
4 answers
4
8.2k views
Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let $T(n)$ be the number of comparisons required to sort $n$ elements. Then $T(n) \leq 2T(n/5) + n$ $T(n) \leq T(n/5) + T(4n/5) + n$ $T(n) \leq 2T(4n/5) + n$ $T(n) \leq 2T(n/2) + n$
asked Sep 12, 2014 in Algorithms Kathleen 8.2k views
...