If f(1)=1,f(2n)=f(n)&f(2n+1)=[f(n)]²-2 for n=1,2,3 ..., Then value of f(1)+f(2)+.....+f(25) is equal to

To find the value of \( f(1) + f(2) + \ldots + f(25) \) given the recursive definitions of the function \( f \), we first need to compute the values of \( f(n) \) for \( n = 1 \) to \( 25 \). The function is defined as follows:

• \( f(1) = 1 \)
• \( f(2n) = f(n) \)
• \( f(2n + 1) = [f(n)]^2 - 2 \)

Let's calculate \( f(n) \) for \( n = 1 \) to \( 25 \) step by step:

Calculating Values of f(n)

We start with the base case:

• \( f(1) = 1 \)

Next, we compute \( f(2) \):

• \( f(2) = f(1) = 1 \)

Now, for \( f(3) \):

• \( f(3) = [f(1)]^2 - 2 = 1^2 - 2 = -1 \)

Continuing with \( f(4) \):

• \( f(4) = f(2) = 1 \)

Next, we find \( f(5) \):

• \( f(5) = [f(2)]^2 - 2 = 1^2 - 2 = -1 \)

For \( f(6) \):

• \( f(6) = f(3) = -1 \)

Now, \( f(7) \):

• \( f(7) = [f(3)]^2 - 2 = (-1)^2 - 2 = 1 - 2 = -1 \)

Next, we compute \( f(8) \):

• \( f(8) = f(4) = 1 \)

Continuing this process, we find:

• \( f(9) = [f(4)]^2 - 2 = 1^2 - 2 = -1 \)
• \( f(10) = f(5) = -1 \)
• \( f(11) = [f(5)]^2 - 2 = (-1)^2 - 2 = -1 \)
• \( f(12) = f(6) = -1 \)
• \( f(13) = [f(6)]^2 - 2 = (-1)^2 - 2 = -1 \)
• \( f(14) = f(7) = -1 \)
• \( f(15) = [f(7)]^2 - 2 = (-1)^2 - 2 = -1 \)
• \( f(16) = f(8) = 1 \) \
• \( f(17) = [f(8)]^2 - 2 = 1^2 - 2 = -1 \) \
• \( f(18) = f(9) = -1 \) \
• \( f(19) = [f(9)]^2 - 2 = (-1)^2 - 2 = -1 \) \
• \( f(20) = f(10) = -1 \) \
• \( f(21) = [f(10)]^2 - 2 = (-1)^2 - 2 = -1 \) \
• \( f(22) = f(11) = -1 \) \
• \( f(23) = [f(11)]^2 - 2 = (-1)^2 - 2 = -1 \) \
• \( f(24) = f(12) = -1 \) \
• \( f(25) = [f(12)]^2 - 2 = (-1)^2 - 2 = -1 \) \

Summing Up the Values

Now that we have calculated the values of \( f(n) \) from \( n = 1 \) to \( n = 25 \), we can summarize them:

• \( f(1) = 1 \)
• \( f(2) = 1 \)
• \( f(3) = -1 \)
• \( f(4) = 1 \)
• \( f(5) = -1 \)
• \( f(6) = -1 \)
• \( f(7) = -1 \)
• \( f(8) = 1 \)
• \( f(9) = -1 \)
• \( f(10) = -1 \)
• \( f(11) = -1 \)
• \( f(12) = -1 \)
• \( f(13) = -1 \)
• \( f(14) = -1 \)
• \( f(15) = -1 \)
• \( f(16) = 1 \)
• \( f(17) = -1 \)
• \( f(18) = -1 \)
• \( f(19) = -1 \)
• \( f(20) = -1 \)
• \( f(21) = -1 \)