Question

Suppose for each positive integer n, an is an integer such that a1 = 1 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a simple expression involving n that evaluates an for each positive integer n. Prove that your guess works for each n ≥ 1.

Suppose for each positive integer n, an is an integer such that a1 = 7 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a simple expression involving n that evaluates an for each positive integer n. Prove that your guess works for each n ≥ 1.

Answer 1

Here given a​​​​​​1=1 and a​​​​​​k=2ak-1+1 ........eq(1) for each integer k​​​​​​​​​​​≥2 .........eq(2)

n is positive integer.

Put k=2,3,4.... We get,

a​​​​​​2 = 2a2-1+1 = 2a1+1 = 2×1+1 = 2+1 = 3

(​​​given a​​​​​​1​​​=1)

Similarly,

a​​​​​​3=2a2+1=2×3+1=7

a​​​​​​4​​​=2a3+1=2×7+1=15

So on..

Now we guess an​​​​​ expression involving n by the help of above expression which evaluate a​​​​​​n for each positive integer i.e. n≥1.

Put k=n+1 in equation (2).

n+1≥2

=> n≥2-1

=> n≥1

See here if we put k= n+1 we get an expression that evaluates for each positive integer.

Put k=n+1 in equation (1), we get

a​​​​​​n+1=2an+1-1+1=2an+1

a​​​​​​n+1=2an+1 is an expression.

Proof: put n=1,2,3,4....we get,

a​​​​​​1+1 = 2a1+1 = 2×1+1 = 3 (a​​​​​​1=1 given)

a​​​​​​​​​2=3

Similarly,

a​​​​​​3=2a2+1=2×3+1=7

a​​​​​​4​​​=2a3+1=2×7+1=15 hence proved.