1 What is the 148th shape 2 How many squares are among the f

1. What is the 148th shape?

2. How many squares are among the first 148 shapes?

3. If we list the numbers below the triangles in order, what kind of sequence will we get? Be as specific as possible.

4. What number will be below the 999th triangle? Explain the reason for your answer.

5. What number will be below the 9999th square? Explain the reason for your answer.

4. In the sequence of the shapes below, the pattern of a square, a triangle, two circles, and another square repeats forever. 1234S6TS910 11 12 13 14 16 16 17 18 192 (0) (5 points) What is the 148th shape? (i) (5 points) How many squares are among the first 148 shapes? (ii) (5 points) If we list the numbers below the triangles in order, what kind of sequence will we get? Be as specific as possible. (iv) (5 points) What number will be below the 999th triangle? Explain the reason for your answer. (v) (10 points) What number will be below the 9999th square? Explain the reason for your answer.

Solution

i) The 148th will be a circle. Every multiple of 5 you get a square. so 5,10,15,20.....145 will be a square. Therefore 146-square, 147-triangle and 148 - circle.

ii) Every 5 numbers has 2 squares. so 145 = 29 * 5. It means 5 comes 29 times. Therefore till 145 we have 29 * 2 = 58 squares. But 146 is also a square. Therfore total squares = 58 + 1 = 59 squares

iii) The 1st triangle is in the 2nd place, the 2nd triangle in the 7th, 3rd in the 12...and so on. so the positions are 2,7,12,17..... This is an Arithmetic Progression (AP), as the difference between any 2 consecutive numbers is always the same-- 7 - 2 = 12 - 7 = 17 - 12 .... = 5

iv) The nth number of an AP = a + (n-1)d. Here n = 999. Therefore 2 + (999-1)*5 = 2 + 998*5 = 4992

(Simple check. What is the number under the 2nd triangle ? it is 7. so 2 + (2 - 1) * 5 = 2 + 5 = 7

v) Difference between every 2 squares = 4

Every 5 number has 2 squares, every 10th number has 4 squares, so every 100th number has 40 squares, every 1000th number has 400 squares.

Therefore 400*25 = 10,000th square has the 25*100 = 25,000 number. Therefore the 9999th square = 25,000-4 = 24,996