Jake arranged 3 beads and 3 sticks to form Pattern 1. Then he added 3 more beads and 6 sticks to form Pattern 2. He continued to form patterns in the same way.

1. How many beads will he need to form Pattern 8?
2. How many sticks will he need to form Pattern 9? 

(a)
Pattern for the number of beads
Pattern 1: 1 + 2 = 3
Pattern 2: 1 + 2 + 3 = 6
Pattern 3: 1 + 2 + 3 + 4 = 10

Formula:
Number of beads
= 1 + 2 + 3 + ... + (Pattern Number + 1)
= ^{(Pattern Number +1) x (Pattern Number + 2)}₂

Number of beads for Pattern 8
= 1 + 2 + 3 + ... + (8 + 1)
= 1 + 2 + 3 + ... + 9
= ^{9 x 10}₂
= 45

(b)
Pattern for the number of sticks
Pattern 1: 3 x (1) = 3
Pattern 2: 3 x (1 + 2) = 9
Pattern 3: 3 x (1 + 2 + 3) = 18

Formula:
Number of sticks
= 3 x (1 + 2 + 3 + ... + Pattern number)
= 3 x ^{Pattern Number x (Pattern Number + 1)}₂

Number of sticks for Pattern 9
= 3 x (1 + 2 + 3 +  ... + 9)
= 3 x (^{9 x 10}₂)
= 135

Answer(s): (a) 45; (b) 135