Page 20 - Ganit Kaushal

Can you write the fifth term of the sequence of hexagonal numbers?

Yes, you got it right, it is: 37 + 4 × 6 = 61

The sequence of hexagonal numbers can also be given by the formula for its n-th term T as,
n
T = 3n(n – 1) + 1; n = 1, 2, 3 … (The explanation of this formula is beyond the scope of this book.)
n
1.6 RELATION BETWEEN NUMBER SEQUENCES

Now it will be interesting to learn some amazing relations between the number sequences. Here we present a
few examples for your observation.

Ex 1. Adding up the sequence of odd numbers gives the sequence of squares. Why? Can this be explained
with the help of a picture? (NCERT)

Sol. First, start adding up the sequence of odd numbers and observe the following beautiful pattern:
1 = 1 (Square of 1)

1 + 3 = 4 (Square of 2)
1 + 3 + 5 = 9 (Square of 3)

1 + 3 + 5 + 7 = 16 (Square of 4)
1 + 3 + 5 + 7 + 9 = 25 (Square of 5)

1 + 3 + 5 + 7 + 9 + 11 = 36 (Square of 6) and so on.
Observe that, if we start adding up the sequence of odd numbers, we get the sequence of squares.

 Amazing! Isn’t it?
Can you say whether this pattern will continue forever and why?

You got it right; the answer is ‘yes’, the pattern does continue forever. We explain
this with the help of the given picture. In the picture, it is shown with the help of
red lines that we can partition the dots in a square grid into an odd number of dots:
1, 3, 5, 7 …… That is why, when we sum up the odd numbers, we get a square.

It is evident from the adjacent picture that
1 + 3 + 5 + 7 + 9 + 11 = 36 (Square of 6).

It is clear from the picture that the sum of odd numbers
always completes a square. Since such a picture can be Note
made for a square of any size, adding up odd numbers Students will learn this formula in
their next classes; for now we just use
gives square numbers and this can continue forever. it without bothering about its origin.
It can also be explained mathematically, as we know that

the formula for the square of the sum of two numbers gives
2
(n + 1) 2 = (n ) + (2n + 1) n = 1, 2, 3, ……
(Square of a natural number) (Square of the previous number) (Next odd number)
Ex 2. Adding up and down a sequence of counting numbers gives the sequence of square numbers. Why?
Explain it with the help of a picture. (NCERT)

18 Mathematics-6