maths 40 marks

Let f(n) be the number of dots in the nth pattern
f(n) = f(n - 1) + 2(n - 1), where n > 1
f(1) = 1
To know the number of dots in the 4th pattern, we can do the bottom-up approach. Progressively, we write
f(2) = f(1) + 2(2 - 1) = 1 + 2 = 3
f(3) = f(2) + 2(3 - 1) = 3 + 4 = 7
f(4) = f(3) + 2(4 - 1) = 7 + 6 = 13
To know the general solution for f(n), for n > 1, we can do the top-down approach,
f(n)
= f(n - 1) + 2(n - 1)
= f(n - 2) + 2(n - 2) + 2(n - 1)
= f(n - 3) + 2 [ (n - 3) + (n - 2) + (n - 1) ]
= ... = f(1) + 2 [ 1 + 2 + 3 + ... + (n - 1) ]
= 1 + 2 [ (n - 1) * n / 2 ]
= 1 + (n - 1) * n
We can check that f(4) = 1 + (4 - 1) * 4 = 13

The 1st pattern has 1 dot
The 2nd pattern has 3 dots, 2(n-1) + 1
The 3rd pattern has 7 dots, 2(n-1) + 3
The 4th pattern has 13 dots, 2(n-1) + 7