294 MATHEMATICS
File Name : C:\Computer Station\Maths-IX\Chapter\Appendix\Appendix– 1 (03– 01– 2006).PM65
Example 5 :Consider the following pattern of numbers called the Pascal’ s Triangle:
Line Sum of numbers
1 1 1
2 1 1 2
3 1 2 1 4
4 1 3 3 1 8
5 1 4 6 4 1 16
6 1 5 10 10 5 1 32
7 : : :
8 : : :What can you conjecture about the sum of the numbers in Lines 7 and 8? What
about the sum of the numbers in Line 21? Do you see a pattern? Make a guess about
a formula for the sum of the numbers in line n.
Solution :Sum of the numbers in Line 7 = 2 × 32 = 64 = 2^6
Sum of the numbers in Line 8 = 2 × 64 = 128 = 2^7
Sum of the numbers in Line 21 = 2^20
Sum of the numbers in Line n = 2n – 1
Example 6 : Consider the so-called triangular numbers Tn:
Fig. A1.1The dots here are arranged in such a way that they form a triangle. Here T 1 = 1,
T 2 = 3, T 3 = 6, T 4 = 10, and so on. Can you guess what T 5 is? What about T 6? What
about Tn?