Problem description:
1. Observe the figure composed of small cubes with the side length of 1: As shown in the figure 1 * *, there are 1 small cubes, of which 1 is visible and 0 is invisible; As shown in Figure 2: * * * has 8 small cubes, of which 7 are visible and 1 is invisible; As shown in Figure 3: * * * has 27 small cubes, of which 19 is visible and 8 are invisible ... Then in Figure 6, there are _ _ _ _ invisible small cubes.
2.( 1) 3 squared-1=8=8 times 1
Square of 5-square of 3 = 16=8 times 2.
Square of 7-Square of 5 =24=8 times 3.
Square of 9-square of 7 =32=8 times 4.
....
15 squared-13 squared = _ _ _ _ _ = _ _ _
(2) the square of (2n+1)-the square of (2n-1)) = _ _ _ _
(3) Calculation: the square of11-99 = _ _ _
3. Observe:
28=5 squared +3 3 1=5 squared +6.
The square of 53 = 2809 09 = 3 56 = 3136 = 6.
(1) after inductive observation of the above formula, it is found that _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(2) Verify _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
4.( 1) Compare the size of two numbers in the following two groups by calculation.
1 square __2 1 square _ _ 3 square __4 square, 4 square __5 square, 5 square _ _ 5 square, 5 square __6 square. .....
(2) From the result of the 1 question, we can guess the size relationship between the (N+ 1) power of n and the (N+ 1) power of n (n is greater than or equal to 3) _ _ _ _ _ _.
(3) According to the following inductive conjecture, draw a general conclusion and try to compare the following two numbers.
1999 is the power of 1998
Analysis:
1. Observe the figure composed of small cubes with the side length of 1: As shown in the figure 1 * *, there are 1 small cubes, of which 1 is visible and 0 is invisible; As shown in Figure 2: * * * has 8 small cubes, of which 7 are visible and 1 is invisible; As shown in Figure 3: * * * has 27 cubes, of which 19 is visible and 8 are invisible ... Then in Figure 6, there is 125 _ _.
2.( 1) 3 squared-1=8=8 times 1
Square of 5-square of 3 = 16=8 times 2.
Square of 7-Square of 5 =24=8 times 3.
Square of 9-square of 7 =32=8 times 4.
....
15 square-13 square = _ 64 _ = _ _ 8 times 8 _ _
(2) the square of (2n+1)-the square of (2n-1) = _ 8n _ _ _
(3) Calculation: the square of11-99 = _ _ _
3. Observe:
28=5 squared +3 3 1=5 squared +6.
The square of 53 = 2809 09 = 3 56 = 3136 = 6.
(1) after inductive observation of the above formula, it is found that _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(2) Verify _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
4.( 1) Compare the size of two numbers in the following two groups by calculation.
65438' s square +0 < 1 < 3' s square and 3' s 4 > 4' s 3, 4' s 5 > 5' s 4, 5' s 6 > 6' s 5. .....
(2) From the result of the question 1, we can guess the relationship between the (N+ 1) power of n and the (N+ 1) power of n (n ≥ 3)_(N+ 1) power > (n ≥ 3).
(3) According to the following inductive conjecture, draw a general conclusion and try to compare the following two numbers.
1999 1998 > 1998 1999.