Question 51 of 51advanced🔍 AnalyzeLong Answer5 marks

Suppose W1=1W_1 = 1, W2=2W_2 = 2 and for n>2n > 2, Wn=W1+W2+⋯+Wn−2+2W_n = W_1 + W_2 + \cdots + W_{n-2} + 2. Find the values of W1W_1, W2W_2, ..., W8W_8. Do you recognise this sequence?

Correct Answer

1,2,3,5,8,13,21,341, 2, 3, 5, 8, 13, 21, 34; Fibonacci sequence

Exercise: END-OF-CHAPTER EXERCISES | Q: 15 | (Page 23)
For More Understanding

Explanation

The student must use the given recursive formula to calculate each term sequentially up to W8W_8. By substituting the previously calculated terms into the formula Wn=W1+W2+⋯+Wn−2+2W_n = W_1 + W_2 + \cdots + W_{n-2} + 2, the sequence emerges.

Observing the pattern, each term from the third term onwards equals the sum of the two preceding terms (e.g., W3=W1+W2W_3 = W_1 + W_2, W4=W2+W3W_4 = W_2 + W_3), which is the defining characteristic of the Fibonacci sequence.

Solution Steps

  1. Step 1: Calculate W3=W1+2=1+2=3W_3 = W_1 + 2 = 1 + 2 = 3

  2. Step 2: Calculate W4=W1+W2+2=1+2+2=5W_4 = W_1 + W_2 + 2 = 1 + 2 + 2 = 5

  3. Step 3: Calculate W5=W1+W2+W3+2=1+2+3+2=8W_5 = W_1 + W_2 + W_3 + 2 = 1 + 2 + 3 + 2 = 8

  4. Step 4: Calculate W6=W1+W2+W3+W4+2=1+2+3+5+2=13W_6 = W_1 + W_2 + W_3 + W_4 + 2 = 1 + 2 + 3 + 5 + 2 = 13

  5. Step 5: Calculate W7=W1+W2+W3+W4+W5+2=1+2+3+5+8+2=21W_7 = W_1 + W_2 + W_3 + W_4 + W_5 + 2 = 1 + 2 + 3 + 5 + 8 + 2 = 21

  6. Step 6: Calculate W8=W1+W2+W3+W4+W5+W6+2=1+2+3+5+8+13+2=34W_8 = W_1 + W_2 + W_3 + W_4 + W_5 + W_6 + 2 = 1 + 2 + 3 + 5 + 8 + 13 + 2 = 34

  7. Step 7: Identify the sequence as the Fibonacci sequence since each term is the sum of the two preceding terms

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