Q2 of 95 Page 68

In the multiplication table made earlier, take a square of nine numbers, instead of four, and mark the numbers at the four corners:


What if we take a square of sixteen numbers?

Now, if we take a square of 16 numbers, then the general terms will be:


Diagonal sums: yx + (y+3)(x+3) = yx + y(x+3) + 3(x+3) = yx +yx +3y +3x +9 = 2yx +3y+3x+9


Y(x+3) + (y+3)x = yx +3y + yx +3x = 2yx + 3y + 3x


Clearly, the difference between the diagonal sums is equal to 9.


More from this chapter

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2

In the multiplication table made earlier, take a square of nine numbers, instead of four, and mark the numbers at the four corners:


Explain using algebra, why this difference is the same for all such squares.

 2

In the multiplication table made earlier, take a square of nine numbers, instead of four, and mark the numbers at the four corners:


What if we take a square of sixteen numbers?

 3

Look at these:

1 × 4 = (2 × 3) – 2


2 × 5 = (3 × 4) – 2


3 × 6 = (4 × 5) – 2


4 × 7 = (5 × 6) – 2


i) Write the next two lines in this pattern.


ii) If we take four consecutive natural numbers, what is the relation between the products of the first and the last, and the product of the middle two?


iii) Write this as a general principle in algebra and explain it.

 3

Look at these:

1 × 4 = (2 × 3) – 2


2 × 5 = (3 × 4) – 2


3 × 6 = (4 × 5) – 2


4 × 7 = (5 × 6) – 2


i) Write the next two lines in this pattern.


ii) If we take four consecutive natural numbers, what is the relation between the products of the first and the last, and the product of the middle two?


iii) Write this as a general principle in algebra and explain it.