Answer:
Please check the explanation.
Explanation:
1) Writing the equation of the pattern
Figure 1 has 4 circles.
Figure 2 has 7 circles.
Figure 3 has 10 circles.
Thus, the sequence becomes
4, 7, 10,...
Finding the common difference between the adjacent terms
d = 7 - 4 = 3, d = 10 - 7 = 3
As the common difference is the same, so this represents the Arithmetic sequence.
so
d = 3
as
a₁ = 4
Thus, the nth term of the Arithmetic sequnce is
aₙ = a₁ + (n-1)d
aₙ = 4 + (n-1)3
aₙ = 4 + 3n - 3
aₙ = 1 + 3n
2) Determining the circles in the 23rd figure
As the nth term is given by
aₙ = 1 + 3n
putting n = 23
a₂₃ = 1 + 3(23)
a₂₃ = 1 + 69
a₂₃ = 70
3) Determining the figure having 82 circles
As the nth term is given by
aₙ = 1 + 3n
Given
aₙ = 82
To determine
n=?
Thus,
aₙ = 1 + 3n
82 = 1 + 3n
3n = 82-1
3n = 81
Dividing both sides by 3
n = 27
Therefore, 27th figure will have 82 circles.