Answer:
a) Expression for the number of tiles in the nth pattern
5 + 2n
b) The pattern number of the pattern made from 21 tiles
8
Step-by-step explanation:
Looking at the figure we see that the pattern has 5 horizontal tiles for all patterns
The number of vertical tiles is 2, 4, 6...
So total number of tiles follows the pattern
5 + 2, 5 + 4, 5 + 6...
7, 9, 11
This is an arithmetic sequence with common difference d = 2(diff. between successive numbers) and initial term a₁ = 7
The nth term of such an arithmetic sequence is given by the formula
aₙ = a₁ + d(n - 1)
a) So, plugging in the values, the expression for the number of tiles in the nth pattern is
aₙ = 7 + 2(n - 1)
which can be simplified as
7+ 2n - 2
aₙ = 5 + 2n Equation (1)
where n = 1, 2, 3,...
Answer a)
5 + 2n
b) Let the unknown pattern number be n
We know this nth pattern has 21 tiles
Substituting for aₙ = 21, d = 2 and a₁= 7 in Equation (1) we get
21 = 5 + 2n
→ 21 - 5 = 5 - 5 + 2n (by subtracting 5 on both sides)
→ 16 = 2n (simplify left and right sides)
→ 2n = 16 (switch sides)
→ 2n/2 = 16/2 (divide by 2 both sides)
→ n = 8
So the pattern number of the pattern made from 21 tiles is 8
