Write a simple formula that generates the below mentioned terms. a) 1,2,2,3,4,4,5,6,6,7,8,8,... b) 1,10,11, 100, 101, 110,111,1000,1001,1010,1011

a.) 1,2,2,3,4,4,5,6,6,7,8,8,....

General formula $a(n)=a(n-1)+a(n-3)-a(n-4)\ \ \ \text{For n>4}$

b.) 1,10,11,100,101,110,111,1000,1001,1010,1011

Yes, the given sequence numbers are represented in binary number format. In the binary number system to represent number, we use only 0’s and 1’s.

The positive integers go like this: 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110……..

Here explicit formula is the digits of a binary number represent powers of 2.

Examples:

1.) $(1100)_2 = (1\times2^3 )+ (1\times2^2) +(0\times2^1) +(0\times2^0)= 8 + 4 +0+0= (12)_{10}= 12$

  2.) $(1101)_2 = (1\times2^3) + (1\times2^2) +(0\times2^1) +(1\times2^0)= 8 + 4 +0+1= (13)_{10}=13$

                    and so on....

Therefore the sequence is : 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110,1111……..