Question:
Observe the following pattern
22 − 12 = 2 + 1
32 − 22 = 3 + 2
42 − 32 = 4 + 3
52 − 42 = 5 + 4
and find the value of
(i) 1002 − 992
(ii) 1112 − 1092
(iii) 992 − 962
Solution:
From the pattern, we can say that the difference between the squares of two consecutive numbers is the sum of the numbers itself.
In a formula:
$(n+1)^{2}-(n)^{2}=(n+1)+n$
Using this formula, we get:
(i) 1002 − 992 = (99 + 1) + 99
= 199
(ii) 1112 − 1092 = 1112 − 1102 + 1102 − 1092
= (111 + 110) + (110 + 109)
= 440
(iii) 992 − 962 = 992 − 982 + 982 − 972 + 972 − 962
= 99 + 98 + 98 + 97 + 97 + 96
= 585