Page 22 - Ganit Kaushal

Can we now affirm that the sum of the first five hexagonal numbers will give us the cube of 5 and so
on?

The answer is Yes! We can show it by a pictorial presentation, similar to the one shown above for the
sum of the first four hexagonal numbers.
Ex 4. Adding up pairs of consecutive triangular numbers gives square numbers. Why? Can it be explained
with the help of a picture? Will it happen forever? (NCERT)
Sol. The answer is certainly ‘yes’. First, observe the following beautiful pattern.

1 + 3 = 4 (Square of 2)
3 + 6 = 9 (Square of 3) (21 dots)
6 + 10 = 16 (Square of 4)

10 + 15 = 25 (Square of 5)
(15 dots)
15 + 21 = 36 (Square of 6)
Let us explain it with the help of the adjoining figure. Here, it is shown that a square consisting of 36 dots
can be divided into two triangular parts (by the red arrow), one consisting of 15 dots and the other consisting
of 21 dots. Both these numbers are consecutive triangular numbers. That is why the sum of two consecutive
triangular numbers gives a square. Since it can be done with a square of any size, this pattern continues forever.

Ex 5. Can you explain the pattern when you start to add up powers of two, starting with 1? Can it be explained
with the help of some formula for the nth term T of this sequence? (NCERT)
n
Sol. Let’s try to answer this question.
First, add up powers of 2, starting with 1 and observe the following:
1 = 1

1 + 2 = 3
1 + 2 + 4 = 7

1 + 2 + 4 + 8 = 15
1 + 2 + 4 + 8 + 16 = 31 and so on
The sequence obtained by adding up powers of two is 1, 3, 7, 15, 31, ……
By observing this sequence carefully, we conclude that it starts with 1 and after that, every term of the sequence
is one more than twice the previous term. So, we can write the sequence {T } with the help of the following
n
formula:

T = 1; T = 2T n – 1 + 1, n = 2, 3, 4, 5, ……
n
1
Explanation of the Formula

We have T = 1 (first term, it’s given)
1
Put n = 2, we get T = 2T + 1 = 2 × 1 + 1 = 3 (second term)
1
2
Put n = 3, we get T = 2T + 1 = 2 × 3 + 1 = 7 (third term)
2
3
Put n = 4, we get T = 2T + 1 = 2 × 7 + 1 = 15 (fourth term)
3
4
Put n = 5, we get T = 2T + 1 = 2 × 15 + 1 = 31 (fifth term) and so on.
5
4
Therefore, the sequence is 1, 3, 7, 15, 31, ……
20 Mathematics-6