0580 P41 - Jun 2013 - Q10 - 14 marks
15871
Stars are made of lines and dots in this pattern.
| Star 1 | Star 2 | Star 3 | Star 4 | Star 5 | Star 7 | Star n | |
|---|---|---|---|---|---|---|---|
| Lines | 10 | 20 | 30 | 40 | … | … | … |
| Dots | 11 | 21 | 31 | 41 | … | … | … |
(a) Complete the table. [4]
The sums of dots in stars 1 and 2, 2 and 3, 3 and 4 are 32, 52, 72.
(b(i)) Find the sum for Star 10 and Star 11. [1]
(b(ii)) Find the sum for Star n and Star (n + 1). [1]
(b(iii)) Find the sum for Star (n + 7) and Star (n + 8). [1]
The total dots in the first n stars is \(5n^2+6n\).
(c(i)) Show this is correct for n = 3. [2]
(c(ii)) Find the total for the first 10 stars. [1]
(d) Add \(5n^2+6n\) and \(10(n+1)+1\) to show the total for the first (n + 1) stars is \(5(n+1)^2+6(n+1)\). Show each step. [4]
