(a) Complete the number pattern by writing out the values of a, b, c and d.
| Pattern | Sum, S | (S + 2) |
| 2 | 2 | 4 |
| 2, 3, 2 | 7 | 9 |
| 2, 3, 4, 3, 2 | a | c |
| 2, 3, 4, 5, 4, 3, 2 | b | d |
(b) What sort of numbers do you have in the (S + 2) column?
(c) Find the sum of 2, 3, 4, ......, 99, 100, 99, ......, 4, 3, 2.
(d) If the middle number of the pattern of positive numbers is M, write a formula relating S and M.
(e) Find the sum of -2, -3, -4, ..., -66, -67, -66, ..., -4, -3, -2.
(f) If 2 + 3 + 4 + 5 + ... + (k - 1) + k + (k - 1) + ... + 5 + 4 + 3 + 2 = 223, find the value of k.
*************************
Answer:
(a)
| Pattern | Sum, S | (S + 2) |
| 2 | 2 | 4 |
| 2, 3, 2 | 7 | 9 |
| 2, 3, 4, 3, 2 | 14 | 16 |
| 2, 3, 4, 5, 4, 3, 2 | 23 | 25 |
(b) Perfect squares
(c) Sum = 100² - 2 = 9998
(d) S = M² - 2
(e) sum of -2, -3, -4, ..., -66, -67, -66, ..., -4, -3, -2
= negative of sum of 2, 3, 4, ..., 66, 67, 66, ..., 4, 3, 2
= - (67² - 2)
= -4487
(f) k² - 2 = 223
k² = 225
k = 15
