Context: We saw earlier how to express the term of the sequence of multiples of , where 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 term for multiples of ? Or, what is the 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 odd number.

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Question 19

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.

Solution

We can find patterns in number sequences.

Step 1 — Finding the nth even number

Let us look at multiples of 2. These are also called even numbers. The first even number is 2. The second even number is 4. The third even number is 6. We can see a clear pattern here. Each even number is its position multiplied by 2. Let us call the position 'n'. So, the $n^{\text{th}}$ even number is $n$ multiplied by 2.

$n^{\text{th}} \text{ even number} = n \times 2$ $= 2n$

$\boxed{\text{The } n^{\text{th}} \text{ even number is } \mathbf{2n}}$

Step 2 — Finding the 100th even number

We want to find the 100th even number. We use our formula from Step 1. Here, the position 'n' is 100. We multiply 100 by 2.

$100^{\text{th}} \text{ even number} = 2 \times 100$ $= 200$

$\boxed{\text{The } 100^{\text{th}} \text{ even number is } \mathbf{200}}$

Step 3 — Finding the 100th odd number

Let us look at odd numbers. The first odd number is 1. The second odd number is 3. The third odd number is 5. We can compare odd and even numbers. The first odd number (1) is one less than the first even number (2). The second odd number (3) is one less than the second even number (4). This pattern continues for all odd numbers. So, the 100th odd number will be one less than the 100th even number. We found the 100th even number in Step 2. It was 200. Now, we subtract 1 from 200.

$100^{\text{th}} \text{ odd number} = 100^{\text{th}} \text{ even number} - 1$ $= 200 - 1$ $= 199$

$\boxed{\text{The } 100^{\text{th}} \text{ odd number is } \mathbf{199}}$

Step 4 — Formula for the nth odd number

We want a general formula for the $n^{\text{th}}$ odd number. From Step 3, we know the $n^{\text{th}}$ odd number is one less than the $n^{\text{th}}$ even number. In Step 1, we found the $n^{\text{th}}$ even number is $2n$ . So, we subtract 1 from $2n$ .

$n^{\text{th}} \text{ odd number} = n^{\text{th}} \text{ even number} - 1$ $= 2n - 1$

$\boxed{\text{The } n^{\text{th}} \text{ odd number is } \mathbf{2n - 1}}$

Answer

(i) The $n^{\text{th}}$ term for multiples of 2 (or $n^{\text{th}}$ even number) is $\mathbf{2n}$ . (ii) The 100th odd number is 199. (iii) The 100th even number is 200. (iv) The formula to find the $n^{\text{th}}$ odd number is $\mathbf{2n - 1}$ .

More questions in IT

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

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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

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