In the sequence of shapes below the pattern of 2 circles 3 s

In the sequence of shapes below, the pattern of 2 circles, 3 squares and 1 triangle repeats forever.

1. What is the 130th shape?

2. How many squares are among the first 129 shapes?

3. What number will be below the 300th square?

2. In the sequence of the shapes below, the pattern of two circles, three squares and one triangle repeats forever. 12345678910 11 12 13 14 15 16 17 18 (i) (3 points) What is the 130th shape? (i) (3 points) How many squares are among the first 129 shapes? (ii) (3 points) What number will be below the 300th square?

Solution

(i) Every 6th number is a triangle...so 6,12,18,24.....126 will be a triangle...127 is a circle, 128 is a circle, 129 is a square and 130 will also be a square.

(ii) Every 6 numbers has 3 squares. 126 is 21 * 6. Therefore upto number 126 we have 21 * 3 = 63 squares. numbers 129 is also a square. Therefore Total = 63 + 1 = 64 squares.

(iii) The squares come under the numbers (3,4,5),(9,10,11),(15,16,17)...therefore 300 squares will come after 100 sets of 3 numbers. Lets look at the last number of each set, as the 300th square will be the last number of the 100th set. It is 5 , 11 , 17...which is an arithmetic progression. The 100 the number of this sequence is given by:    t_n = a + (n -1 )*d, where a is the 1st number = 3, d is the common difference = 6 and n is the no. of terms = 100.

Therefore, t_n = 3 + (100-1)*6 = 3 + 594 = 597. The 300th square is the 597th number.