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 recognise this sequence?

Step-by-step solution

Idea: Note that the sum stops at W_n−2, two places back. Work out the values, then compare W_n with W_n−1.

1. W₃ = W₁ + 2 = 1 + 2 = 3; W₄ = 1 + 2 + 2 = 5; W₅ = 1 + 2 + 3 + 2 = 8.1 mark
2. W₆ = 1 + 2 + 3 + 5 + 2 = 13; W₇ = 1 + 2 + 3 + 5 + 8 + 2 = 21; W₈ = 1 + 2 + 3 + 5 + 8 + 13 + 2 = 34.1 mark
3. Why: for n ≥ 4, W_n = (W₁ + … + W_n−3 + 2) + W_n−2 = W_n−1 + W_n−2. Also W₃ = 3 = W₂ + W₁.1 mark
4. So each term is the sum of the previous two, starting 1, 2: this is the Virahānka–Fibonacci sequence 1, 2, 3, 5, 8, 13, 21, 34, ….1 mark

Check: Each value from W₃ on equals the sum of the two before it: 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Adding up to W_n−1 instead of W_n−2 (that gives 1, 2, 5, 10, …).
• Forgetting the + 2 at the end.
• Calling it an AP or a GP. The differences (1, 1, 2, 3, 5, …) and the ratios both change.

How this can come in the exam

Try one yourself

Starting from 2, 3 and adding the previous two terms each time, write the first eight terms.

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• 1 = 1, 4 = 1 + 3, 9 = 1 + 3 + 5, 16 = 1 + 3 + 5 + 7, and so on. Each term in the square number sequence is the sum of the odd numbers up to that term. This interesting relationship between the odd numbers and square numbers can be represented by the diagram in the figure. Can you explain the relationship?
• Consider the sequence 1, 4, 7, 10, 13, … Can you predict the next four terms? Can you derive the first 10 terms of the sequence obtained by adding all the terms up to a given term of this sequence? (Hint: The first term is 1. The second term is 1 + 4 = 5, the third term is 1 + 4 + 7 = 12, and so on.)

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