Question -
Answer -
Let P (n):┬а1/2 + 1/4 + 1/8 + тАж + 1/2n┬а=1 тАУ 1/2n
Let us check for n = 1,
P (1): 1/21┬а= 1 тАУ 1/21
: 1/2 = 1/2
P (n) is true for n = 1.
Now, let us check for P (n) is true for n = k, and have toprove that P (k + 1) is true.
Let P (k):┬а1/2 + 1/4 + 1/8 + тАж + 1/2k┬а=1 тАУ 1/2k┬атАж (i)
So,
1/2 + 1/4 + 1/8 + тАж + 1/2k┬а+ 1/2k+1
Now, substituting the value of P (k) we get,
= 1 тАУ 1/2k┬а+ 1/2k+1┬аby usingequation (i)
= 1 тАУ ((2-1)/2k+1)
P (n) is true for n = k + 1
Hence, P (n) is true for all n тИИ N.