Solve the questions based off the pattern

Question

1. b. How many squares would be in the 10th image?
2. c. How many squares would be in the nth image?
3. d. What is the rule for this pattern in terms of n?
4. e. Is the rule arithmetic, geometric, or neither? Explain.

Answer 1

Let a_n = number of squares in case n

a₁ = 6 = (2)(3)

a₂ = 12 = (3)(4)

a₃ = 20 = (4)(5)..... a_n = (n+1)(n+2)

So, a₁₀ = (11)(12) = 132

The sequence is not arithmetic since 12 - 6 is not the same as 20 - 12.

The sequence is not geometric because 12/6 is different from 20/12

Answer 2

In each case one row and one column is added, so case 1 has 1 + 1 row and 1 +2 columns. The number os squares is (1+1) (1+2) = 6. The second case has 2+1 rows and 2+2 columns leading to 3x4 =12 squares. The third case has 3+1 rows and 3+2 columns or 4x5 squares.

The 10th case would have 10+1 rows and 10+2 columns or 11 x 12 squares.

The nth image would have (n+1) (n+2) squares which is the rule for the nth image.

It seems to be neither strictly arithmetic nor geometric, but I am not entirely certain of this.