`1/2+1/4+1/8+1/16+.......+1/2^n=`

Options

A. `1-1/2^n`
B. `1-1/3^n`
C. `1-1/4^n`
D. `1-1/5^n`

Answer

A

Explanation

`" Let "P(n) : 1/2+1/4+1/8+.........+1/2^(n)=1-1/2^(n)`
for n=1
`L.H.S. =1/2`
`R.H.S. =1-1/2^(1)`
`=1-1/2=1/2`
`:. " "L.H.S. =R.H.S.`
therefore P (n) is true for n=1.
Let P (n) true for n=k
`P (n) : 1/2+1/4+1/8+......+1/2^(k) =1-1/2^(k)`
for n=K+1
`P(k+1) : 1/2+1/4+1/8+......+1/2^(k)+1/2^(k+1) =1-1/2^(k)+1/2^(k+1)`
`=1-((1)/(2^(k))-(1)/(2^(k+1)))`
`=1-((2-1)/(2^(k+1)))=1-(1)/(2^(K+1))`
`rArr` P (n) is also true for n=K+1
Hence from the principle of mathematical induction the given statement is true for all natural numbers n.