Consider the following arrays of squares Form a conjecture t

Consider the following arrays of squares Form a conjecture that will give you the number of squares in the n^th array of squares Attempt to form a proof for this conjecture.

Solution

2.

No. of squares in 1st array = 1

No. of squares in 2nd array = 1+2 = 3

No. of squares in 3rd array = 1+2+3 = 6

No. of squares in 4th array = 1+2+3+4 = 10

Thus, no. of squares in nth array is given by = 1+2+3+ .... + n = n(n+1)/2

Thus, from the formula for no. of squares in nth array i.e. n(n+1)/2, we can verify for n=1, 2, 3 & 4

For n = 1, No. of squares = 1*2/2 = 1

For n = 2, No. of squares = 2*3/2 = 3

For n = 3, No. of squares = 3*4/2 = 6

For n = 4, No. of squares = 4*5/2 = 10

Thus, from above it is shown that the no. of squares in nth array is given by n(n+1)/2