12 Exercises

#$&*

course mth151

Please don't write me off yet. i am on a death march project at work, so i have fallen behind. i will catch up by the end of this week. thank you in advance

007. Triangular, Square, Pentagonal Numbers

*********************************************

Question: `q001. Note that there are 7 questions in this assignment.

Sketch three points A, B and C forming an equilateral triangle on a piece of paper, with point A at the lower left-hand corner, point B at the lower right-hand corner and point C at the top. Sketch the segments AB and AC. Now double the lengths of AB and AC, and place a point at each of the endpoints of these segments. Connect these new endpoints to form a new equilateral triangle. Two sides of this triangle will have three points marked while the new side will only have its two endpoints marked. Fix that by marking that middle point, so all three sides of your new triangle are marked the same. How many marked points were there in the original triangle, and how many are there in the new triangle?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Points in original triangle: 3

Points in new triangle: 6

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q002. Extend the two sides that meet at A by distances equal to the original lengths AC and AB and mark the endpoints of the newly extended segments. Each of the newly extended sides will have 4 marked points. Now connect the new endpoints to form a new right triangle. Mark points along the new side at the same intervals that occur on the other two sides. How many marked points are on your new triangle, and how many in the whole figure?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

points in new triante: 6

points in whole figure:10

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q003. Continue the process for another step-extend each side by a distance equal to the original point-to-point distance. How many points do you have in the new triangle?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 13 points total

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q004. Continue the process for one more step. How many points do you have in the new triangle?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 15 points

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q005. The sequence of marked points is 3, 6, 10, 15, 21. What do expect will be the next number in this sequence?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 28

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q006. How can you tell, in terms of the process you used to construct these triangles, that the next number should be 7 greater?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: because the previous step broughtthe count of endpoints to 6 and the one before that brought the count to 5. A pattern of increase by +1 has been established.

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q007. How do you know this sequence will continue in this manner?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: All previous steps have shown an increase by +1; This is the established pattern

confidence rating #$&*: very confident.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

=============================================================

007. `Query 7

Question: `qQuery 1.2.6 seq 2, 57, 220, 575, 1230, 2317 ... by successive differences

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Line1: 2, 57, 220, 575, 1230, 2317==>2317_1675=3992

Line2: 55, 163, 355, 655, 1087==>1087+588=1675

Line3: 108, 192, 300, 432==>432+156=588

Line4: 84, 108, 132==> 132+24=156

Line5: 24

Answer = 3992

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q1.2.18 1^2 + 1 = 2^2 - 2; 2^2 + 2 = 3^2 - 3; 3^2 + 3 = etc.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 3^2+3=4^2-4; 4^2+4=5^2+5

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q1.2.30 state in words (1 + 2 + ... + n ) ^ 2 = 1^3 + 2^3 + ... + n^3

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q1.2.36 1 st triangular # div by 3, remainder; then 2d etc. Pattern.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: Triangular numbers are:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 104. When divided by 3 the remainders are

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0. Using inductive reasoning, you can see that every third number is a multiple of 3, so the pattern will continue in the same way

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q1.2.48 use formula to find the 12 th octagonal number. Explain in detail how you used the formula to find this number.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: List of first 8 octagonal numbers= 1, 8, 21, 40, 65, 96, 133, 176

The nth octagonal number can be found by the following:

n(3n - 2)

12(3*12-2)

12(36-2)

12(34)=408

408 is the 12th octagonal number

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

"

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

12 Exercises

#$&*

course mth151

Please don't write me off yet. i am on a death march project at work, so i have fallen behind. i will catch up by the end of this week. thank you in advance

007. Triangular, Square, Pentagonal Numbers

*********************************************

Question: `q001. Note that there are 7 questions in this assignment.

Sketch three points A, B and C forming an equilateral triangle on a piece of paper, with point A at the lower left-hand corner, point B at the lower right-hand corner and point C at the top. Sketch the segments AB and AC. Now double the lengths of AB and AC, and place a point at each of the endpoints of these segments. Connect these new endpoints to form a new equilateral triangle. Two sides of this triangle will have three points marked while the new side will only have its two endpoints marked. Fix that by marking that middle point, so all three sides of your new triangle are marked the same. How many marked points were there in the original triangle, and how many are there in the new triangle?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Points in original triangle: 3

Points in new triangle: 6

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q002. Extend the two sides that meet at A by distances equal to the original lengths AC and AB and mark the endpoints of the newly extended segments. Each of the newly extended sides will have 4 marked points. Now connect the new endpoints to form a new right triangle. Mark points along the new side at the same intervals that occur on the other two sides. How many marked points are on your new triangle, and how many in the whole figure?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

points in new triante: 6

points in whole figure:10

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q003. Continue the process for another step-extend each side by a distance equal to the original point-to-point distance. How many points do you have in the new triangle?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 13 points total

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q004. Continue the process for one more step. How many points do you have in the new triangle?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 15 points

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q005. The sequence of marked points is 3, 6, 10, 15, 21. What do expect will be the next number in this sequence?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 28

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q006. How can you tell, in terms of the process you used to construct these triangles, that the next number should be 7 greater?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: because the previous step broughtthe count of endpoints to 6 and the one before that brought the count to 5. A pattern of increase by +1 has been established.

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q007. How do you know this sequence will continue in this manner?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: All previous steps have shown an increase by +1; This is the established pattern

confidence rating #$&*: very confident.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

=============================================================

007. `Query 7

Question: `qQuery 1.2.6 seq 2, 57, 220, 575, 1230, 2317 ... by successive differences

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Line1: 2, 57, 220, 575, 1230, 2317==>2317_1675=3992

Line2: 55, 163, 355, 655, 1087==>1087+588=1675

Line3: 108, 192, 300, 432==>432+156=588

Line4: 84, 108, 132==> 132+24=156

Line5: 24

Answer = 3992

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q1.2.18 1^2 + 1 = 2^2 - 2; 2^2 + 2 = 3^2 - 3; 3^2 + 3 = etc.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: 3^2+3=4^2-4; 4^2+4=5^2+5

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q1.2.30 state in words (1 + 2 + ... + n ) ^ 2 = 1^3 + 2^3 + ... + n^3

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q1.2.36 1 st triangular # div by 3, remainder; then 2d etc. Pattern.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: Triangular numbers are:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 104. When divided by 3 the remainders are

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0. Using inductive reasoning, you can see that every third number is a multiple of 3, so the pattern will continue in the same way

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q1.2.48 use formula to find the 12 th octagonal number. Explain in detail how you used the formula to find this number.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution: List of first 8 octagonal numbers= 1, 8, 21, 40, 65, 96, 133, 176

The nth octagonal number can be found by the following:

n(3n - 2)

12(3*12-2)

12(36-2)

12(34)=408

408 is the 12th octagonal number

confidence rating #$&*: very confident

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

"

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

#*&!

@& Be sure to see my notes on preceding submissions.*@