Suppose W1 = 1, W2 = 2 and for n > 2, Wn = W1 + W2 + ... + Wn-2 + 2. Find the values of W1, W2,..., W8. Class 9

Suppose W₁ = 1, W₂ = 2 and for n > 2, W_n = W₁ + W₂ + ... + W_n-2 + 2. Find the values of W₁, W₂,..., W₈. Class 9

Question 1.

Suppose W₁ = 1, W₂ = 2 and for n > 2, W_n = W₁ + W₂ + ... + W_n-2 + 2. Find the values of W₁, W₂,..., W₈. Do you recognize this sequence? Class 9

Solution:

Given,

W₁ = 1, W₂ = 2 and

for n > 2, W_n = W₁ + W₂ + ... + W_n-2 + 2

W₃ = W₁ + 2 = 1 + 2 = 3

W₄ = W₁ + W₂ + 2 = 1 + 2 + 2 = 5

W₅ = W₁ + W₂ + W₃ + 2 = 1 + 2 + 3 + 2 = 8

W₆ = W₁ + W₂ + W₃ + W₄ + 2 = 1 + 2 + 3 + 5 + 2 = 13

W₆ = W₁ + W₂ + W₃ + W₄ + W₅ + 2 = 1 + 2 + 3 + 5 + 8 + 2 = 21

W₈ = W₁ + W₂ + W₃ + W₄ + W₅ + W₆ + 2 = 1 + 2 + 3 + 5 + 8 + 13 + 2 = 34

∵ Each term = sum of the two previous terms, 1, 2, 3, 5, 8, 13, 21, 34, ....

∴ This is the Virahanka - Fibonacci sequence

W_n = W_n-1 + W_n-2 for n ≥ 3