a)

Number of circles in Pattern 1 = 1

Number of circles in Pattern 2 = 3
= 1 + 2

Number of circles in Pattern 3 = 6
= 1 + 2 + 3

Number of circles in Pattern 4 = 10
= 1 + 2 + 3 + 4

Number of circles in Pattern 5 = 15
= 1 + 2 + 3 + 4 + 5

So, number of circles in pattern 6
= 1 + 2 + 3 + 4 + 5 + 6
= 21




b)

Number of dots in Pattern 1 = 4

Number of dots in Pattern 2 = 8
= 4 + 4

Number of dots in Pattern 3 = 13
= 4 + 4 + 5

Number of dots in Pattern 4 = 19
= 4 + 4 + 5 + 6

Number of dots in Pattern 5 = 26
= 4 + 4 + 5 + 6 + 7


So, number of dots in Pattern 6
= 4 + 4 + 5 + 6 + 7 + 8
= 34




c)

Following the above pattern,

Number of circles in pattern 101= 1 + 2 + 3 + 4 +..... + 97 + 98 + 99 + 100 + 101

Notice that :

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
4 + 97 = 101
.
.
.
.
48 + 53 = 101
49 + 52 = 101
50 + 51 = 101

So from 1 to 100, there are 50 pairs of numbers. And each pair adds up to 101.

So 101 x 50 = 5050

We have a 101 that is not paired.

So the total number of circles for Pattern 101

= 5050 + 101
= 5151

Very good solution, J.

Thanks Arnold for the kind compliment!