Answer:
See explanation
Step-by-step explanation:
1 step:
n=1, then
![\sum \limits_{j=1}^1 2^j=2^1=2\\ \\2(2^1-1)=2(2-1)=2\cdot 1=2](https://tex.z-dn.net/?f=%5Csum%20%5Climits_%7Bj%3D1%7D%5E1%202%5Ej%3D2%5E1%3D2%5C%5C%20%5C%5C2%282%5E1-1%29%3D2%282-1%29%3D2%5Ccdot%201%3D2)
So, for j=1 this statement is true
2 step:
Assume that for n=k the following statement is true
![\sum \limits_{j=1}^k2^j=2(2^k-1)](https://tex.z-dn.net/?f=%5Csum%20%5Climits_%7Bj%3D1%7D%5Ek2%5Ej%3D2%282%5Ek-1%29)
3 step:
Check for n=k+1 whether the statement
![\sum \limits_{j=1}^{k+1}2^j=2(2^{k+1}-1)](https://tex.z-dn.net/?f=%5Csum%20%5Climits_%7Bj%3D1%7D%5E%7Bk%2B1%7D2%5Ej%3D2%282%5E%7Bk%2B1%7D-1%29)
is true.
Start with the left side:
![\sum \limits _{j=1}^{k+1}2^j=\sum \limits _{j=1}^k2^j+2^{k+1}\ \ (\ast)](https://tex.z-dn.net/?f=%5Csum%20%5Climits%20_%7Bj%3D1%7D%5E%7Bk%2B1%7D2%5Ej%3D%5Csum%20%5Climits%20_%7Bj%3D1%7D%5Ek2%5Ej%2B2%5E%7Bk%2B1%7D%5C%20%5C%20%28%5Cast%29)
According to the 2nd step,
![\sum \limits_{j=1}^k2^j=2(2^k-1)](https://tex.z-dn.net/?f=%5Csum%20%5Climits_%7Bj%3D1%7D%5Ek2%5Ej%3D2%282%5Ek-1%29)
Substitute it into the ![\ast](https://tex.z-dn.net/?f=%5Cast)
![\sum \limits _{j=1}^{k+1}2^j=\sum \limits _{j=1}^k2^j+2^{k+1}=2(2^k-1)+2^{k+1}=2^{k+1}-2+2^{k+1}=2\cdot 2^{k+1}-2=2^{k+2}-2=2(2^{k+1}-1)](https://tex.z-dn.net/?f=%5Csum%20%5Climits%20_%7Bj%3D1%7D%5E%7Bk%2B1%7D2%5Ej%3D%5Csum%20%5Climits%20_%7Bj%3D1%7D%5Ek2%5Ej%2B2%5E%7Bk%2B1%7D%3D2%282%5Ek-1%29%2B2%5E%7Bk%2B1%7D%3D2%5E%7Bk%2B1%7D-2%2B2%5E%7Bk%2B1%7D%3D2%5Ccdot%202%5E%7Bk%2B1%7D-2%3D2%5E%7Bk%2B2%7D-2%3D2%282%5E%7Bk%2B1%7D-1%29)
So, you have proved the initial statement
Your questions are:
1.1. 96 points for 6 games
1.2. 592 students for 17 classrooms
2.1. 15 yd² for $2 per square yard
2.2. 26 ounce at $0.15 per ounce
The first set of problem, that is 1.1. and 1.2 are answered by dividing the numbers.
1.1. 16 points per game
1.2. 34.8 or 35 students per classroom
For the second set, we just have to multiply to get the cost of each.
2.1. $30
2.2. $3.9
Because 3/4 is equal to 75% and 3/8 is equal to .325% .75% is closer to 100% than .325% is.
Answer:
b= -1
Step-by-step explanation:
Step 1: Find out value of x
x-2 = 0
x= 2
Step 2: Substitute value of x in the equation
x^3 + bx^2- 4 = 0
(2)^3 + b (2)^2 - 4 = 0
8 + 4b - 4=0
4 - 4b = 0
4b = -4
b = -1
2x-4/x-5 = 0 (Determine the defined range)
x \neq 5
2x-4 = 0 (Move constant to the right side and change its sign)
2x=4
x=2, x \neq 5 (Check if the solution is in the defined range)
x=2