I do not understand a problem from a text book or how to solve the problem for the answer. Could shomeone show me the steps (show work) on how to solve this question.
The sum of the intergers from 1 through n is n(n+1)/2. the sum of the squares of the intergers from 1 through n is n(n+1)(2n+1)/6. The sum of the cubes of the integers from 1 through n is n squared(n+1)squared/4. Use the appropriate expressions to find the following values.
(A). The sum of the integers from 1 through 30.
(B) The sum of the squares of the integers from 1 through 30
(C) The sum of the cubes of integers from 1 through 30.
(D) The square of the sum of the integers from 1 though 30.
(E) The cube of the sum of integers from 1 through 30.
I's sure its much easier than what I am making it out to be. I tried working the problems but am lost in what it' asking.
They have given you the formulas for each of the question for A, B, and C
I will do B
Sum of squares from 1 to n
= n(n+1)(2n+1)/6
so the sum of the squares from 1 to 30, n=30
just plug it in
sum of squares = 30(31)(61)/6 = 9455
do A and C the same way
Once you have the answer for A,
square that result for D
cube the result for A to get E
During a walk, Dave completed the first two miles at a pace of 3mph. He walked the next two miles at a pace of 4.5 mph. Would his walk take more, less or the samr time if he walked the entire distance of four miles at a steady pace of 3.5 mph?
Sure, let's go step by step to solve each part of the question.
(A) The sum of the integers from 1 through 30:
To find the sum of integers from 1 through n, we can use the formula n(n+1)/2. In this case, n = 30. Substituting the value into the formula, we get:
Sum = 30(30+1)/2
Simplifying the expression, we have:
Sum = 30(31)/2
Sum = 930
Therefore, the sum of integers from 1 through 30 is 930.
(B) The sum of the squares of the integers from 1 through 30:
To find the sum of the squares of integers from 1 through n, we use the formula n(n+1)(2n+1)/6. Again, n = 30. Substituting the value into the formula, we get:
Sum of Squares = 30(30+1)(2(30)+1)/6
Simplifying the expression, we have:
Sum of Squares = 30(31)(61)/6
Sum of Squares = 30(961)/6
Sum of Squares = 30(160.17)
Sum of Squares ≈ 4805.1
Therefore, the sum of the squares of integers from 1 through 30 is approximately 4805.1.
(C) The sum of the cubes of integers from 1 through 30:
To find the sum of the cubes of integers from 1 through n, we use the formula n^2(n+1)^2/4. Once again, n = 30. Substituting the value into the formula, we get:
Sum of Cubes = 30^2(30+1)^2/4
Simplifying the expression, we have:
Sum of Cubes = 30^2(31)^2/4
Sum of Cubes = 900(961)/4
Sum of Cubes = 864900/4
Sum of Cubes = 216225
Therefore, the sum of the cubes of integers from 1 through 30 is 216225.
(D) The square of the sum of integers from 1 through 30:
To find the square of the sum of integers from 1 through n, we first need to find the sum using the formula n(n+1)/2. Again, n = 30. Substituting the value into the formula, we get:
Sum = 30(30+1)/2
Sum = 930
Now, we can square the sum:
Square of Sum = (930)^2
Square of Sum = 864900
Therefore, the square of the sum of integers from 1 through 30 is 864900.
(E) The cube of the sum of integers from 1 through 30:
To find the cube of the sum of integers from 1 through n, we first need to find the sum using the formula n(n+1)/2. Once again, n = 30. Substituting the value into the formula, we get:
Sum = 30(30+1)/2
Sum = 930
Now, we can cube the sum:
Cube of Sum = (930)^3
Cube of Sum = 804357000
Therefore, the cube of the sum of integers from 1 through 30 is 804357000.
I hope this explanation helps you solve the problem! Let me know if you have any further questions.