What is the parity of the 20th term of the Virahāṅka sequence?

Question

Homi · Accepted Answer

We need to find the parity. This is for the 20th Virahāṅka term.

Step 1 — Understanding the Virahāṅka Sequence

The Virahāṅka sequence is also called the Fibonacci sequence. Each term is the sum of the two previous terms. Let us list the first few terms. The first term is 1. The second term is 1. Let $F_n$ be the $n$ -th term. So, $F_1 = 1$ . And $F_2 = 1$ . Then $F_3 = F_2 + F_1$ .

$F_3 = 1 + 1$

$= 2$

Then $F_4 = F_3 + F_2$ .

$F_4 = 2 + 1$

$= 3$

Then $F_5 = F_4 + F_3$ .

$F_5 = 3 + 2$

$= 5$

Then $F_6 = F_5 + F_4$ .

$F_6 = 5 + 3$

$= 8$

Then $F_7 = F_6 + F_5$ .

$F_7 = 8 + 5$

$= 13$

Then $F_8 = F_7 + F_6$ .

$F_8 = 13 + 8$

$= 21$

Then $F_9 = F_8 + F_7$ .

$F_9 = 21 + 13$

$= 34$

Then $F_{10} = F_9 + F_8$ .

$F_{10} = 34 + 21$

$\boxed{55}$

Diagram 1

Step 2 — Finding the Parity Pattern

We will check each term's parity. Is it Odd (O) or Even (E)? Let us write down the parity for each term. The first term $F_1$ is 1. Its parity is Odd. The second term $F_2$ is 1. Its parity is Odd. The third term $F_3$ is 2. Its parity is Even. The fourth term $F_4$ is 3. Its parity is Odd. The fifth term $F_5$ is 5. Its parity is Odd. The sixth term $F_6$ is 8. Its parity is Even. The seventh term $F_7$ is 13. Its parity is Odd. The eighth term $F_8$ is 21. Its parity is Odd. The ninth term $F_9$ is 34. Its parity is Even. The tenth term $F_{10}$ is 55. Its parity is Odd.

The parity sequence is O, O, E, O, O, E. It continues this way. We can see a repeating pattern here. The pattern is O, O, E. This pattern repeats every 3 terms.

$\boxed{\text{The parity pattern is O, O, E}}$

Step 3 — Determining the Parity of the 20th Term

The pattern of parities repeats every 3 terms. We need to find the parity of the 20th term. We divide 20 by 3. This shows its place in the pattern.

$20 \div 3$

$= 6 \text{ with a remainder of } 2$

The O, O, E pattern repeats 6 full times. Then we look at the remainder. The remainder is 2. The 20th term has parity like the second term. This is from the pattern. The second pattern term is O (Odd). So, the 20th term will be Odd.

$\boxed{\text{The 20th term is Odd}}$

Answer

(i) The parity of the 20th term of the Virahāṅka sequence is Odd.

Question 24

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