1. Simplify 1/2! + 2/3! + 3/4! ... + 11/12!
2.let x=2^100,y^60 and z=10^30.What is the smallest number among the three?
can you check my answers?
1. 1
2. y=3^60
#1
sum(1) = 1/2
sum(2) = 1/2 + 2/3!
= 1/2 + 1/3 = 5/6
sum(3) = 5/6 + 3/4! = ( (3+1)! - )/(3+1)!
= 5/6 + 1/8 = 23/24
sum(4) = 23/24 + 4/5!
= 23/24 + 1/30 = 119/120 = ((4+1)! - 1)/(4+1)!
it looks like
sum(n) = ( (n+1)! - 1)/(n+1)!
so sum(11) = (12! - 1)/12! = 479001599/479001600 which is not yet 1, but sure is close to 1
#2 is correct, did you use your calculator?
My solution is too long. I didn't use calculator
for #2, I suggest taking logs
x = 2^100
logx = 100log2 = 30.102..
y = 3^60
logy = 60log3 = 28.627..
z = 10^30
logz = 30log10 = 30
thus logy < logz < logx
then y < z < x ----> y is the smallest
To simplify the expression 1/2! + 2/3! + 3/4! ... + 11/12!, we can break it down and evaluate each term individually.
First, let's start with the term 1/2!. The exclamation mark (!) represents the factorial operation. So, 2! is equal to 2 * 1, which is 2. Now, we have 1/2.
Moving on to the next term, 2/3!. Here, 3! is equal to 3 * 2 * 1, which is 6. So, we have 2/6, which simplifies to 1/3.
Now, let's evaluate the next term, 3/4!. 4! is equal to 4 * 3 * 2 * 1, which is 24. So, we have 3/24, which simplifies to 1/8.
Continuing this pattern, the next term, 4/5!, simplifies to 1/30.
This process continues until we reach the last term, 11/12!. 12! is a large number, but we can use the previously defined pattern to simplify the expression.
After evaluating all the terms, we get the simplified expression: 1/2 + 1/3 + 1/8 + 1/30 + ...
As for the second question, let's evaluate the values of x, y, and z.
Given x = 2^100, we can calculate its value using the exponentiation rule. 2^100 equals a very large number.
Moving on to y = 2^60, it seems like there is an error in the equation you provided. Are you referring to a different variable instead of y?
Lastly, z = 10^30. We know that any number raised to the power of 0 is 1. Therefore, 10^30 is equal to 1 followed by 30 zeros.
From these calculations, it seems that x is the largest number, followed by z. Without the correct equation for y, we cannot determine its value or compare it with the other two numbers.