A recursive rule or formula does not only have to involve the previous term — it could involve the previous two or more terms. The most famous example of such a sequence is V1, V2, V3, … where V1 = 1, V2 = 2, and Vn = Vn−1 + Vn−2 for n ≥ 3. So we get the sequence 1, 2, 3, 5, 8, 13, 21, 34, … where each term is obtained by adding the previous two. Can you write the next two terms of this sequence?
Answer: 55 and 89 (21 + 34 = 55, 34 + 55 = 89). This is the Virahānka–Fibonacci sequence.
Step-by-step solution
Idea: Each new term is the sum of the two terms just before it.
- V9 = V8 + V7 = 34 + 21 = 55.1 mark
- V10 = V9 + V8 = 55 + 34 = 89.1 mark
The next two terms are 55 and 89.
Check: Next would be 89 + 55 = 144; every term is the sum of the two before ✓.
Answer to write in the exam
V9 = 34 + 21 = 55
V10 = 55 + 34 = 89
∴ Next two terms: 55, 89
Common mistakes that cost marks
- Adding the last term to itself (34 + 34 = 68). Add the last two terms.
- Adding the first two terms again (1 + 2) instead of the last two.
- Adding the last term and the one two places back: 34 + 13 = 47. Add the last two terms: 21 + 34 = 55.
How this can come in the exam
MCQ (1 mark)
In 2, 5, 7, 12, 19, …, each term is the sum of the previous two. The next term is
- 26
- 31
- 38
- 24
Show answer
(B) 31
12 + 19 = 31.
Try one yourself
A sequence starts 3, 4 and each term is the sum of the previous two. Write its first seven terms.
Show answer
3, 4, 7, 11, 18, 29, 47.
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