HOMEWORK

Homework - 4

Homework - 4

CMATH 6108

Question 1

Solution

10 ! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10

10 ! = 3628800

Therefore, there are 3628800 options for planning the layout of the walkway being built using the rectangular pavers.

Question 2

Solution

= 1! x 2! x 3! x 3! x 2! x 1!

= 1 x 2 x 6 x 6 x 2 x 1

= 144

The Super Sum

Question 1

Solution

The two different strategies to compute the sum of the consecutive numbers from 1 - 100 have been mentioned below;

i) Using the Gauss' formula for finding sum of a series, which has been mentioned below;

S = n ( n + 1) / 2

ii) Using the formula of arithmetic series, which has been mentioned below;

S = n / 2 (2a + (n-1) d)

Where, S = sum of progression, n = number of terms, a = value of first term, d = difference between two terms

Question 2

Solution

a) n = 99, a = 1, d = 1

Using formula, S = n / 2 * (2a + (n - 1) d)

S = 99 / 2 * (2 + (99 - 1) 1)

S = 99 / 2 * (100)

S = 49.5 * 100

S = 4950

Now, using formula, S = n (n + 1) / 2

S = 99 (99 + 1) / 2

S = 9900 / 2

S = 4950

b) n = 101, a = 1, d = 1

Using formula, S = n / 2 * (2a + (n - 1) d)

S = 101 / 2 * (2 + (101 - 1) 1)

S = 101 / 2 * (102)

S = 50.5 * 102

S = 5151

Now, using formula, S = n (n + 1) / 2

S = 101 (101 + 1) / 2

S = 10302 / 2

S = 5151

c) n = 500, a = 1, d = 1

Using formula, S = n / 2 * (2a + (n - 1) d)

S = 500 / 2 * (2 + (500 - 1) 1)

S = 250 * (501)

S = 250 * 501

S = 125250

Now, using formula, S = n (n + 1) / 2

S = 500 (500 + 1) / 2

S = 250500 / 2

S = 125250

d) n = 175, a = 1, d = 1

Using formula, S = n / 2 * (2a + (n - 1) d)

S = 175 / 2 * (2 + (175 - 1) 1)

S = 175 / 2 * (176)

S = 87.5 * 176

S = 15400

Now, using formula, S = n (n + 1) / 2

S = 175 (175 + 1) / 2

S = 30800 / 2

S = 15400

Question 3

i) n is even

S = (n + 1) / 2 * (2a + (n - 1) d)

ii) n is odd

S = n / 2 * (2a + (n - 1) d)

Question 4

Solution

a) n = 200, a = 2, d = 2

Using formula, S = n / 2 * (2a + (n - 1) d)

S = 200 / 2 * (4 + (200 - 1) 2)

S = 200 ...