Number Play | IT

Question 38

Write the next 3 numbers in the sequence: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \underline{\quad}, \underline{\quad}, \underline{\quad}, \dots$

If you have to write one more number in the sequence above, can you tell whether it will be an odd number or an even number (without adding the two previous numbers)?

Solution

We will find the pattern in the sequence and use it to calculate the next numbers and their parity.

Step 1 — Finding the pattern

Let us look at the given numbers in the sequence. The numbers are $1, 2, 3, 5, 8, 13, 21, 34, 55, 89$ . Let us check if there is a rule to get the next number. We can try adding the previous two numbers.

First number is $1$ . Second number is $2$ . Third number: $1 + 2 = \mathbf{3}$ . This matches. Fourth number: $2 + 3 = \mathbf{5}$ . This matches. Fifth number: $3 + 5 = \mathbf{8}$ . This matches. Sixth number: $5 + 8 = \mathbf{13}$ . This matches. Seventh number: $8 + 13 = \mathbf{21}$ . This matches. Eighth number: $13 + 21 = \mathbf{34}$ . This matches. Ninth number: $21 + 34 = \mathbf{55}$ . This matches. Tenth number: $34 + 55 = \mathbf{89}$ . This matches.

The pattern is that each number is the sum of the two numbers before it. This is called the Fibonacci sequence.

Step 2 — Calculating the next 3 numbers

We need to find the 11th, 12th, and 13th numbers. The 9th number is 55. The 10th number is 89.

The 11th number will be the sum of the 9th and 10th numbers. $55 + 89$ $= 144$ The first missing number is 144.

The 12th number will be the sum of the 10th and 11th numbers. $89 + 144$ $= 233$ The second missing number is 233.

The 13th number will be the sum of the 11th and 12th numbers. $144 + 233$ $= 377$ The third missing number is 377.

$\boxed{144, 233, 377}$

Step 3 — Determining the parity of the next number

We need to find the parity (odd or even) of the 14th number. We can look at the pattern of odd and even numbers in the sequence. Let 'O' stand for Odd and 'E' stand for Even.

The sequence is: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, \dots$ Let us write down their parities: $1$ is Odd. $2$ is Even. $3$ is Odd. (Odd + Even = Odd) $5$ is Odd. (Even + Odd = Odd) $8$ is Even. (Odd + Odd = Even) $13$ is Odd. (Odd + Even = Odd) $21$ is Odd. (Even + Odd = Odd) $34$ is Even. (Odd + Odd = Even) $55$ is Odd. (Odd + Even = Odd) $89$ is Odd. (Even + Odd = Odd) $144$ is Even. (Odd + Odd = Even) $233$ is Odd. (Odd + Even = Odd) $377$ is Odd. (Even + Odd = Odd)

The pattern of parities is: O, E, O, O, E, O, O, E, O, O, E, O, O, ... We can see a repeating pattern of (O, E, O, O). No, this is not quite right. Let's look at the rule for adding parities: Odd + Odd = Even Odd + Even = Odd Even + Odd = Odd Even + Even = Even

Let $P_n$ be the parity of the $n$ -th number. $P_1 = \text{O}$ $P_2 = \text{E}$ $P_3 = P_1 + P_2 = \text{O} + \text{E} = \text{O}$ $P_4 = P_2 + P_3 = \text{E} + \text{O} = \text{O}$ $P_5 = P_3 + P_4 = \text{O} + \text{O} = \text{E}$ $P_6 = P_4 + P_5 = \text{O} + \text{E} = \text{O}$ $P_7 = P_5 + P_6 = \text{E} + \text{O} = \text{O}$ $P_8 = P_6 + P_7 = \text{O} + \text{O} = \text{E}$

The pattern of parities is (O, E, O, O, E, O, O, E, ...). The repeating block is O, E, O. This sequence repeats every three terms. Let's check: Term 1: O Term 2: E Term 3: O Term 4: O (This is $P_2+P_3 = E+O = O$ ) Term 5: E (This is $P_3+P_4 = O+O = E$ ) Term 6: O (This is $P_4+P_5 = O+E = O$ ) Term 7: O (This is $P_5+P_6 = E+O = O$ ) Term 8: E (This is $P_6+P_7 = O+O = E$ )

The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O, E... The pattern of parities is O, E, O, O, E, O, O, E, O, O, E, O, O, E. The repeating pattern is O, E, O, O, E, O, O, E. No.

Let's re-evaluate the repeating pattern of parities: $F_1$ : O $F_2$ : E $F_3$ : O (O+E) $F_4$ : O (E+O) $F_5$ : E (O+O) $F_6$ : O (O+E) $F_7$ : O (E+O) $F_8$ : E (O+O) $F_9$ : O (O+E) $F_{10}$ : O (E+O) $F_{11}$ : E (O+O) $F_{12}$ : O (O+E) $F_{13}$ : O (E+O)

The parities are: O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O. The pattern is O, E, O, O, E, O, O, E, O, O, E, O, O.

More questions in IT

What do the numbers in the figure below tell us?

Remember the children from the Grade 6 textbook of mathematics? Now, they call out numbers using a different rule.

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The children rearrange themselves and each one says a number based on the new arrangement.

Context: The children rearrange themselves and each one says a number based on the new arrangement.

Q. Could you figure out what these numbers convey? Observe and try to find out.

Write down the number each child should say based on this rule for the arrangement shown below.

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Context: Kishor has some number cards and is working on a puzzle: There are 5 boxes, and each box should contain exactly 1 number card. The numbers in the boxes should sum to 30. Can you help him find a way to do it? Can you figure out which 5 cards add to 30? Is it possible? There are many ways of choosing 5 cards from this collection. Is there a way to find a solution without checking all possibilities? Let us find out.

Q. Add a few even numbers together. What kind of number do you get? Does it matter how many numbers are added?

Context: As we see in the figure, adding any number of even numbers will result in a number which can still be arranged in pairs without any leftovers. In other words, the sum will always be an even number.

Q. Now, add a few odd numbers together. What kind of number do you get? Does it matter how many odd numbers are added?

Context: Can we also think of an odd number as one less than a collection of pairs? This figure shows that the sum of two odd numbers must always be even! This along with the other figures here are more examples of a proof!

Q. What about adding 3 odd numbers? Can the resulting sum be arranged in pairs?

Explore what happens to the sum of: (a) 4 odd numbers (b) 5 odd numbers (c) 6 odd numbers

Q10

Two siblings, Martin and Maria, were born exactly one year apart. Today they are celebrating their birthday. Maria exclaims that the sum of their ages is 112. Is this possible? Why or why not?

Q11

Context: Small Squares in Grids In a $3 \times 3$ grid, there are 9 small squares, which is an odd number. Meanwhile, in a $3 \times 4$ grid, there are 12 small squares, which is an even number.

Q. Given the dimensions of a grid, can you tell the parity of the number of small squares without calculating the product?

Q12

Find the parity of the number of small squares in these grids:

(a) $27 \times 13$

(b) $42 \times 78$

Q13

Context: Consider the algebraic expression: $3n + 4$ . For different values of $n$ , the expression has different parity:

(a) Come up with an expression that always has even parity. Some examples are: $100p$ and $48w - 2$ . Try to find more.

(b) Come up with expressions that always have odd parity.

(d) The expression $6k + 2$ evaluates to $8, 14, 20, \dots$ (for $k = 1, 2, 3, \dots$ ) — many even numbers are missing.

(e) Are there expressions using which we can list all the even numbers? Hint: All even numbers have a factor 2.

(f) Are there expressions using which we can list all odd numbers?

Q19

Context: We saw earlier how to express the $n^{\text{th}}$ term of the sequence of multiples of $4$ , where $n$ is the letter-number that denotes a position in the sequence (e.g., first, twenty third, hundred and seventeenth, etc.).

(1) What would be the $n^{\text{th}}$ term for multiples of $2$ ? Or, what is the $n^{\text{th}}$ even number?

Let us consider odd numbers.

(2) What is the 100th odd number?

To answer this question, consider the following question:

(3) What is the 100th even number?

(4) Write a formula to find the $n^{\text{th}}$ odd number.

Q21

Context: Observe this $3 \times 3$ grid. It is filled following a simple rule — use numbers from $1 - 9$ without repeating any of them. There are circled numbers outside the grid. The numbers in the yellow circles are the sums of the corresponding rows and columns.

Q. Fill the grids below based on the rule mentioned above:

Q22

Make a couple of questions like this on your own and challenge your peers.

Q27

Can 1 occur in a corner position? For example, can it be placed as follows?

Q. If yes, then there should exist three ways of adding 1 with two other numbers to give 15. We have 1 + 5 + 9 = 1 + 6 + 8 = 15. Is any other combination possible?

Q. Similarly, can 9 can be placed in a corner position?

Q29

Can you find the other possible positions for 1 and 9?

Now, we have one full row or column of the magic square! Try completing it!

[Hint: First fill the row or columns containing 1 and 9]

Q30

Choose any magic square that you have made so far using consecutive numbers. If $m$ is the letter-number of the number in the centre, express how other numbers are related to $m$ , how much more or less than $m$ .

[Hint: Remember, how we described a $2 \times 2$ grid of a calendar month in the Algebraic Expressions chapter].

Q31

Context: Choose any magic square that you have made so far using consecutive numbers. If $m$ is the letter-number of the number in the centre, express how other numbers are related to $m$ , how much more or less than $m$ .

[Hint: Remember, how we described a $2 \times 2$ grid of a calendar month in the Algebraic Expressions chapter].

Q. Once the generalised form is obtained, share your observations with the class.

Q32

Chautīsā means 34. Why do you think they called it the Chautīsā Yantra? Every row, column and diagonal in this magic square adds up to 34. Can you find other patterns of four numbers in the square that add up to 34?

Q33

How many rhythms are there with 8 beats consisting of short syllables (1 beat) and long syllables (2 beats)? That is, in how many ways can one fill 8 beats with short and long syllables, where a short syllable takes one beat of time and a long syllable takes two beats of time?

Q34

Context: A short syllable takes one beat of time and a long syllable takes two beats of time. Some possibilities to fill 8 beats are:

long long long long
short short short short short short short short
short long long short long
long long short short long

Q. Can you find others?

Q35

Context: We can write the number 8 as a sum of 1's and 2's in several ways, for example:

8 = 2 + 2 + 2 + 2
8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
8 = 1 + 2 + 2 + 1 + 2
8 = 2 + 2 + 1 + 1 + 2

Q. Do you see other ways?

Q36

Try writing the number 5 as a sum of 1s and 2s in all possible ways in your notebook! How many ways did you find? (You should find 8 different ways!) Can you figure out the answer without listing down all the possibilities? Can you try it for n = 8?

Q37

Context: Thus, there are 8 rhythms having 5 beats! The reason this method works is that every 5-beat rhythm must begin with either a '1+' or a '2+'. If it begins with a '1+', then the remaining numbers must give a 4-beat rhythm, and we can write all those down. If it begins with a 2+, then the remaining number must give a 3-beat rhythm, and we can write all those down. Therefore, the number of 5-beat rhythms is the number of 4-beat rhythms, plus the number of 3-beat rhythms. How many 6-beat rhythms are there? By the same reasoning, it will be the number of 5-beat rhythms plus the number of 4-beat rhythms, i.e., $8 + 5 = 13$ . Thus, there are 13 rhythms having 6 beats.

Q. Use the systematic method to write down all 6-beat rhythms, i.e., write 6 as the sum of 1's and 2's in all possible ways. Did you get 13 ways?

Q38

Write the next 3 numbers in the sequence: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \underline{\quad}, \underline{\quad}, \underline{\quad}, \dots$

If you have to write one more number in the sequence above, can you tell whether it will be an odd number or an even number (without adding the two previous numbers)?

Q39

Context: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots$

What is the parity of each number in the sequence? Do you notice any pattern in the sequence of parities?

Q40

Context: Let us look at one more example. Here $\text{K2}$ means that the number is a 2-digit number having the digit '2' in the units place and 'K' in the tens place. $\text{K2}$ is added to itself to give a 3-digit sum $\text{HMM}$ :

$\begin{array}{r} \text{K2} \\ +\ \text{K2} \\ \hline \text{HMM} \end{array}$

Q. What digit should the letter $\text{M}$ correspond to? Both the tens place and the units place of the sum have the same digit. What about $\text{H}$ ? Can it be 2? Can it be 3?

Q41

Context: These types of questions can be interesting and fun to solve! Here are some more questions like this for you to try out. Find out what each letter stands for. Share how you thought about each question with your classmates; you may find some new approaches.

Q. Find out what each letter stands for:

(i) $\begin{array}{r} \text{YY} \\ +\quad \text{Z} \\ \hline \text{ZOO} \end{array}$

(ii) $\begin{array}{r} \text{B5} \\ +\quad \text{3D} \\ \hline \text{ED5} \end{array}$

(iii) $\begin{array}{r} \text{KP} \\ +\quad \text{KP} \\ \hline \text{PRR} \end{array}$

(iv) $\begin{array}{r} \text{C1} \\ +\quad \text{C} \\ \hline \text{1FF} \end{array}$