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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. $P = Q + 2k$

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

Correct Answer: $A$
Substitute k=3 then we get p=9 and q=12  on verifying we get option A.
by

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
but option c also satisfies given condition........................... checking for each k
@sanjay how option c will true please explain?
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
@SHIV_KANNAUJ

LoL
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...

### 1 comment

Nice approach.Thanks :-)

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

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  :)
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

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

by

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