617 views

Let $\DeclareMathOperator{S}{sgn} \S (x)= \begin{cases} +1 & \text{if } x \geq 0 \\ -1 & \text{if } x < 0 \end{cases}$

What is the value of the following summation?

$$\sum_{i=0}^{50} \S \left ( (2i - 1) (2i - 3) \dots (2i - 99) \right)$$

1. $0$
2. $-1$
3. $+1$
4. $25$
5. $50$

edited by

@srestha Ma'am

I think answer should be -1 .

there are 45 terms in this sequencsequence.So sgn(0) will be -1 .  likewise . even will give -1 and odd will give +1.

now, 0 and 1 will make pair and result in 0.

so forth... 8 and 9 will be paired and result in 0.

48 and 49 will make pair and result in 0.

so we left with 50 which is even so it will give "-1" .

p.s : yes key is +1 but where i am making mistake then ?

@Headshot, for even it will be +1 and for odd it will be -1 check answer below.

$\sum_{i=0}^{50}sgn((2i-1)(2i-3)\ldots(2i-99))$

There are 50(even number) terms present in the product.

Now we need to open summation for $i = 0$ to $50$

$i=0 , sgn((-1)(-3)\ldots(-99)) = +1$

Because we know that there are $50$ terms present in product which is an even number so it will be $+1.$

$i=1, sgn((1)(-1)\ldots (-97)) = -1$

Because in product there are $49$ negative terms and $1$ term is positive, answer will be negative.

From $i=0$ and $i=1,$ we understood that summation is giving $+1$ for even values of $i$ and $-1$ for odd values of $i.$

From $0$ to $50$ there are $26$ even and $25$ odd numbers are present.

So, answer will be $+1.$

first term : 1

last term : 99

difference : 2

$\frac{99-1}{2} + 1$

so there are 45 terms which is an odd number.

i = 0 to 50 , is "i" value , why you considered it as number of terms ?

$\small \frac{99-1}{2} +1 = 45 How?$ should b 50 right. $\small \frac{98}{2}+1 = 49 +1 = 50.$

For second doubt check answer i was not considering i as no. of terms. I have framed correctly now.

Oh..my bad.. calculated no. of terms wrong.

thanx.

s

by