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Please check the explanation.

Step-by-step 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.

right

step-by-step explanation:

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