2)
In the multiplication table made earlier, take a square of nine num-
bers, instead of four, and mark the numbers at the four corners:
9 12 15
(12) 16 (20
i) What is the difference of diagonal sums?
ii) Explain using algebra, why this difference is the same for all such
squares.
iii) What if we take a square of sixteen numbers?
Answers
Answer:
the difference of diagonal sums = 0
Step-by-step explanation:
2 3 4
7 8 9
12 13 14
2 + 14 - 4 + 12
= 16 - 16
= 0
11 12 13
16 17 18
21 22 23
11 + 23 - 21 + 13
= 34 - 34
= 0
the difference of diagonal sums = 0
a a+ 1 a + 2
a+5 a + 6 a + 7
a+ 10 a + 11 a + 12
a + a + 12 - (a + 10 + a + 2)
= 2a + 12 - (2a + 12)
= 0
difference of diagonal sums = 0
if we take a square of sixteen numbers , still the difference of diagonal sums = 0
a a+1 a+2 a+3
a+5 a+6 a+7 a+8
a+10 a+11 a+12 a+13
a+15 a+16 a+17 a+18
a + a + 18 - (a + 15 + a + 3)
= 2a + 18 - (2a + 18)
= 0
the difference of diagonal sums = 0
Question Talks about multiplication table
then Difference of sum of Diagonals
would be d² for (d + 1) squares
2 * 2 - Square (4 numbers)
Difference of sum of Diagonals = 1² = 1
3 * 3 - Square ( 9 numbers)
Difference of sum of Diagonals = 2² = 4
4 * 4 - Square ( 16 numbers)
Difference of sum of Diagonals = 3² = 9
Proof :
an a(n+1) a(n+d)
(a+1)n (a+1)(n+1) (a+1)(n+d)
(a+d)n (a+d)(n+1) (a+d)(n+d)
an + (a+d)(n+d) - ((a+d)n + a(n+d))
= d²
