Question

Given the proposition,

P(n): 1 + 2 + 2^2 + 2^3 + . . . + 2n = 2n+1 - 1, n = 0, 1, 2, . . .

Find the values of:

P(0)

P(1)

P(2)

P(n+1)

Answer 1

Answer and Explanation:

Given :
`P(n): 1 + 2 + 2^2 + 2^3 + . . . + 2^n = 2^(n+1) - 1`, n=0,1,2,..

To find : The values of following expression ?

Solution :

The function is
`P(n)=2^(n+1) - 1`

1) Value of P(0),

`P(0)=2^(0+1) - 1`

`P(0)=2^(1) - 1`

`P(0)=2 - 1`

`P(0)=1`

2) Value of P(1),

`P(1)=2^(1+1) - 1`

`P(1)=2^(2) - 1`

`P(1)=4- 1`

`P(1)=3`

3) Value of P(2),

`P(2)=2^(2+1) - 1`

`P(2)=2^(3) - 1`

`P(2)=8- 1`

`P(2)=7`

4) Value of P(n+1),

`P(n+1)=2^(n+1+1) - 1`

`P(n+1)=2^(n+2) - 1`